Solutions of a selection of , 20, diophantine equations with, 5, variables of degree, 5 By Shalosh B. Ekhad Theorem Number, 1 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 650288293599288729095549347 X[1] 4 - 7968220655492420245379266987 X[1] X[2] 4 - 10106635629484855344337387529 X[1] X[3] 4 - 12793419667444974442176216422 X[1] X[4] 4 + 1358626352835915197635903056 X[1] X[5] 3 2 + 36710629875615010762527515386 X[1] X[2] 3 + 98972297429153283051668251744 X[1] X[2] X[3] 3 + 121489689571739639056564605412 X[1] X[2] X[4] 3 - 13379501162664350956306518912 X[1] X[2] X[5] 3 2 + 67262791330988555189798638114 X[1] X[3] 3 + 160770087471372323720229199928 X[1] X[3] X[4] 3 - 20906264707607239595404315956 X[1] X[3] X[5] 3 2 + 99203904492769417333370480464 X[1] X[4] 3 - 22329105281674810520518188648 X[1] X[4] X[5] 3 2 + 1744130190336888270353496804 X[1] X[5] 2 3 - 78648755981788231003277409794 X[1] X[2] 2 2 - 346017676067178072317332381110 X[1] X[2] X[3] 2 2 - 406113269280504472501163776776 X[1] X[2] X[4] 2 2 + 46434286930614910531207440408 X[1] X[2] X[5] 2 2 - 485191564614289079497782070818 X[1] X[2] X[3] 2 - 1147675829074917859149864974652 X[1] X[2] X[3] X[4] 2 + 148784724960625847975376749292 X[1] X[2] X[3] X[5] 2 2 - 684898771244410005889044513912 X[1] X[2] X[4] 2 + 158581229568701175203125170168 X[1] X[2] X[4] X[5] 2 2 - 12453179111055101317005160092 X[1] X[2] X[5] 2 3 - 249537332498757698599722177430 X[1] X[3] 2 2 - 812174194921593564566401307268 X[1] X[3] X[4] 2 2 + 129908872030586963985744161940 X[1] X[3] X[5] 2 2 - 947220206195983664483666737680 X[1] X[3] X[4] 2 + 257359386876494732019508051584 X[1] X[3] X[4] X[5] 2 2 - 22323176194959485905941730476 X[1] X[3] X[5] 2 3 - 379439544745188256961244608536 X[1] X[4] 2 2 + 134897945130530080052839640424 X[1] X[4] X[5] 2 2 - 21708218595788853472881360072 X[1] X[4] X[5] 2 3 + 1149650438102850566002663656 X[1] X[5] 4 + 77239258276072831452796991747 X[1] X[2] 3 + 508337890693985394031696466272 X[1] X[2] X[3] 3 + 560340030110392741461430691980 X[1] X[2] X[4] 3 - 67776942532653601562333400192 X[1] X[2] X[5] 2 2 + 1111007514179387910079236135090 X[1] X[2] X[3] 2 + 2593599171646718394823171517952 X[1] X[2] X[3] X[4] 2 - 326114054191473939639017590572 X[1] X[2] X[3] X[5] 2 2 + 1477427780099383901994911901168 X[1] X[2] X[4] 2 - 353178732367022708377272046968 X[1] X[2] X[4] X[5] 2 2 + 25878417197502187821249988380 X[1] X[2] X[5] 3 + 1158457788113386166637041734864 X[1] X[2] X[3] 2 + 3783332677041762114644257991772 X[1] X[2] X[3] X[4] 2 - 598283679412357254978058128168 X[1] X[2] X[3] X[5] 2 + 4375232555643147265636184628960 X[1] X[2] X[3] X[4] - 1166880792831094491847996861872 X[1] X[2] X[3] X[4] X[5] 2 + 105631018644358670222751409512 X[1] X[2] X[3] X[5] 3 + 1693248328918060458819239642272 X[1] X[2] X[4] 2 - 614044052363006927226027802368 X[1] X[2] X[4] X[5] 2 + 96736867836744957510666435984 X[1] X[2] X[4] X[5] 3 - 5951205583372065972699625680 X[1] X[2] X[5] 4 + 521827810702325780449389666395 X[1] X[3] 3 + 2028676180454741680119384961544 X[1] X[3] X[4] 3 - 379670989627246891278575298492 X[1] X[3] X[5] 2 2 + 3224198417398524774391137127824 X[1] X[3] X[4] 2 - 1066194569913224110897231886040 X[1] X[3] X[4] X[5] 2 2 + 96925420910445842027107705356 X[1] X[3] X[5] 3 + 2448258925382416621367707211696 X[1] X[3] X[4] 2 - 1033061166707857154768182287888 X[1] X[3] X[4] X[5] 2 + 185530825271919701517618260736 X[1] X[3] X[4] X[5] 3 - 9700222847889529578903445872 X[1] X[3] X[5] 4 + 716273265299121790031343637712 X[1] X[4] 3 - 355018338157124140857339994800 X[1] X[4] X[5] 2 2 + 87266959165544431461076176912 X[1] X[4] X[5] 3 - 9933141282435682893677646000 X[1] X[4] X[5] 4 + 306035311365101291390539632 X[1] X[5] 5 - 28042370629096599957809974547 X[2] 4 - 256690311271486046779479979409 X[2] X[3] 4 - 267384678844175949289663304546 X[2] X[4] 4 + 34091678957940417906697044600 X[2] X[5] 3 2 - 826617863162457111803155018634 X[2] X[3] 3 - 1833333457084264724100330894028 X[2] X[3] X[4] 3 + 233365103387961455641184525988 X[2] X[3] X[5] 3 2 - 987305566704756386772996376808 X[2] X[4] 3 + 246770354182670776261760245416 X[2] X[4] X[5] 3 2 - 17477255786782396238641076868 X[2] X[5] 2 3 - 1250801762403161071931836971598 X[2] X[3] 2 2 - 4227276159829568439327676707624 X[2] X[3] X[4] 2 2 + 608199623211117479034957566796 X[2] X[3] X[5] 2 2 - 4784578081669256821571663783040 X[2] X[3] X[4] 2 + 1244465199749756458477506777552 X[2] X[3] X[4] X[5] 2 2 - 102852575444235064404306189420 X[2] X[3] X[5] 2 3 - 1769556758776900251309783287080 X[2] X[4] 2 2 + 658304167936512307238329020120 X[2] X[4] X[5] 2 2 - 97717143291236495829526214664 X[2] X[4] X[5] 2 3 + 5730987682530815170245362280 X[2] X[5] 4 - 1160479426731936594685396318531 X[2] X[3] 3 - 4501578789670946973170767442548 X[2] X[3] X[4] 3 + 858435347479598051085479484108 X[2] X[3] X[5] 2 2 - 7282395560386772382771618778248 X[2] X[3] X[4] 2 + 2305908122938694492883544411560 X[2] X[3] X[4] X[5] 2 2 - 231039879455598102615223833804 X[2] X[3] X[5] 3 - 5481619823164539068428460979136 X[2] X[3] X[4] 2 + 2249403360151384148786187296544 X[2] X[3] X[4] X[5] 2 - 401850654941355829604176641408 X[2] X[3] X[4] X[5] 3 + 25861206660354501784509022992 X[2] X[3] X[5] 4 - 1549933303476178840104804747856 X[2] X[4] 3 + 777564437443267236196322522160 X[2] X[4] X[5] 2 2 - 184430637885515048921536475472 X[2] X[4] X[5] 3 + 22511197788260063699719589904 X[2] X[4] X[5] 4 - 1008902376549447264940936464 X[2] X[5] 5 - 478375768956127723522361776033 X[3] 4 - 2128952372411134807445984153270 X[3] X[4] 4 + 430633373167423166860586758860 X[3] X[5] 3 2 - 4038859204307203423698382598320 X[3] X[4] 3 + 1559346010811341269851561430240 X[3] X[4] X[5] 3 2 - 141619093739609967209286125316 X[3] X[5] 2 3 - 4205360514972175546534703476216 X[3] X[4] 2 2 + 2124380992466065865595416761032 X[3] X[4] X[5] 2 2 - 405284097393424210572513646776 X[3] X[4] X[5] 2 3 + 20448828543386426028191586216 X[3] X[5] 4 - 2341045210047164875161986451152 X[3] X[4] 3 + 1353581803603807196065660244592 X[3] X[4] X[5] 2 2 - 370150038661032340022133884880 X[3] X[4] X[5] 3 + 42547663139956173628816553424 X[3] X[4] X[5] 4 - 1233359214060434602427813232 X[3] X[5] 5 - 534043905667714468222349462752 X[4] 4 + 343616790452346318779248316832 X[4] X[5] 3 2 - 113838166968611824539273113280 X[4] X[5] 2 3 + 20160947931414603149659938336 X[4] X[5] 4 - 1494474990379176423021746400 X[4] X[5] 5 + 20764757877906305613130272 X[5] + 109834150262780130281291599776 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j 11 t + 6 t - 16 t + 17 t + 15 ) a[1, j] t = ------------------------------------- / 5 4 3 2 ----- -t + 13 t - 15 t + 11 t + 8 t + 1 j = 0 infinity ----- 4 3 2 \ j 13 t + 5 t + 16 t + 19 t - 4 ) a[2, j] t = ------------------------------------- / 5 4 3 2 ----- -t + 13 t - 15 t + 11 t + 8 t + 1 j = 0 infinity ----- 4 3 2 \ j 8 t + 5 t + 4 t - 2 t - 3 ) a[3, j] t = ------------------------------------- / 5 4 3 2 ----- -t + 13 t - 15 t + 11 t + 8 t + 1 j = 0 infinity ----- 4 3 2 \ j -8 t - 12 t - 13 t - 7 t + 8 ) a[4, j] t = ------------------------------------- / 5 4 3 2 ----- -t + 13 t - 15 t + 11 t + 8 t + 1 j = 0 infinity ----- 4 3 2 \ j 18 t - 8 t - 15 t - 20 t - 5 ) a[5, j] t = ------------------------------------- / 5 4 3 2 ----- -t + 13 t - 15 t + 11 t + 8 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 2 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = -4126246300262389121117198443613 X[1] 4 + 29308872136786472102758436140409 X[1] X[2] 4 + 22812049182539813915430154557371 X[1] X[3] 4 + 9022079799505659896011300233184 X[1] X[4] 4 - 9750847251966605084588605426755 X[1] X[5] 3 2 - 40611880094265903042634612487030 X[1] X[2] 3 - 99091913804892272725561478478544 X[1] X[2] X[3] 3 - 73647490284311489577852468647168 X[1] X[2] X[4] 3 + 64957531854669558372010161116100 X[1] X[2] X[5] 3 2 - 42801931836638701302875686223246 X[1] X[3] 3 - 48810117296606917484337706263800 X[1] X[3] X[4] 3 + 41908803572862397066018913353080 X[1] X[3] X[5] 3 2 - 4380873668519905690796404226816 X[1] X[4] 3 + 14278344656967261020822703725556 X[1] X[4] X[5] 3 2 - 5251952053526123514754213748514 X[1] X[5] 2 3 - 37639724767189404240903853085786 X[1] X[2] 2 2 + 67490312584317995417682044337906 X[1] X[2] X[3] 2 2 + 132758611131176841440813308242144 X[1] X[2] X[4] 2 2 - 144902868514990767618506961490854 X[1] X[2] X[5] 2 2 + 112686070012027483848322445631246 X[1] X[2] X[3] 2 + 213368829485580759171028226469528 X[1] X[2] X[3] X[4] 2 - 184680280682065397745987418185156 X[1] X[2] X[3] X[5] 2 2 + 51200037135460622810173657169568 X[1] X[2] X[4] 2 - 81663331322125122572428810042284 X[1] X[2] X[4] X[5] 2 2 + 20853081797453485800531283194606 X[1] X[2] X[5] 2 3 + 34748711959541226473292432514810 X[1] X[3] 2 2 + 78754613252157278891972564102016 X[1] X[3] X[4] 2 2 - 57625359968845101595322587602474 X[1] X[3] X[5] 2 2 + 29451344747163970993515071237712 X[1] X[3] X[4] 2 - 55202449187819739491374877255628 X[1] X[3] X[4] X[5] 2 2 + 12764627686355800066599741094710 X[1] X[3] X[5] 2 3 - 1434518904618434018528369520424 X[1] X[4] 2 2 - 4989618441301600937533409191512 X[1] X[4] X[5] 2 2 + 6927874866330214736248763613876 X[1] X[4] X[5] 2 3 - 391360363126148412148047885630 X[1] X[5] 4 + 98676265458111345856137454177043 X[1] X[2] 3 + 88266408916818162683882595956192 X[1] X[2] X[3] 3 - 42795994062904049054756818152656 X[1] X[2] X[4] 3 + 130541529182351538763421814869532 X[1] X[2] X[5] 2 2 - 39677374300908944198643756193086 X[1] X[2] X[3] 2 - 179038365923139249187238611323768 X[1] X[2] X[3] X[4] 2 + 244474466010690269994643380218472 X[1] X[2] X[3] X[5] 2 2 - 96524396344484528974169336603664 X[1] X[2] X[4] 2 + 136649105927482290447114441465804 X[1] X[2] X[4] X[5] 2 2 - 27036217522170244511909563115250 X[1] X[2] X[5] 3 - 54896264473003232043703248167392 X[1] X[2] X[3] 2 - 168998843626188446855993230620240 X[1] X[2] X[3] X[4] 2 + 148183296020046096691188236916852 X[1] X[2] X[3] X[5] 2 - 130650731821434414852702812571312 X[1] X[2] X[3] X[4] + 184503563157128299969328684843688 X[1] X[2] X[3] X[4] X[5] 2 - 32024469741586936165349589444984 X[1] X[2] X[3] X[5] 3 - 6142906238500512798329613824864 X[1] X[2] X[4] 2 + 24044246763816169500219000771120 X[1] X[2] X[4] X[5] 2 - 19980354221712601680428621687016 X[1] X[2] X[4] X[5] 3 + 1029651451505669756575380322548 X[1] X[2] X[5] 4 - 12583716400311682118877627878581 X[1] X[3] 3 - 45651938271969128163678810516248 X[1] X[3] X[4] 3 + 28930324854149385787706105020248 X[1] X[3] X[5] 2 2 - 42813464731375556780059604683632 X[1] X[3] X[4] 2 + 61231183348083125953380418903932 X[1] X[3] X[4] X[5] 2 2 - 9033858544901753775969106042194 X[1] X[3] X[5] 3 - 2018194181260901545214175958304 X[1] X[3] X[4] 2 + 17259490484524739704502579326992 X[1] X[3] X[4] X[5] 2 - 13615405611827173278448540671792 X[1] X[3] X[4] X[5] 3 + 630989951198275661903760631536 X[1] X[3] X[5] 4 + 937206742954438475766223593536 X[1] X[4] 3 - 179343832324696966992960130032 X[1] X[4] X[5] 2 2 - 1957420172795421618147783262608 X[1] X[4] X[5] 3 + 368395090743940475130723564108 X[1] X[4] X[5] 4 + 4207410592351750445894055423 X[1] X[5] 5 - 45312112038003129781297630190783 X[2] 4 - 77947433666293202751007876013869 X[2] X[3] 4 - 23964090977551277708156855612624 X[2] X[4] 4 - 40421331679484061043207436665287 X[2] X[5] 3 2 - 32442022018886922855590383680218 X[2] X[3] 3 + 1680129148022725407248396461192 X[2] X[3] X[4] 3 - 99235170182468535931851205706460 X[2] X[3] X[5] 3 2 + 34873454903168356531565383682992 X[2] X[4] 3 - 66947209778682765538401125233764 X[2] X[4] X[5] 3 2 + 11585688164238425717944243252638 X[2] X[5] 2 3 + 10028704969111165219231951131298 X[2] X[3] 2 2 + 55516734310158501281153424432672 X[2] X[3] X[4] 2 2 - 88643162960475271568565057473382 X[2] X[3] X[5] 2 2 + 76580802268494979601834051979792 X[2] X[3] X[4] 2 - 131204593584076919597509491073116 X[2] X[3] X[4] X[5] 2 2 + 20089693404108918187210237380102 X[2] X[3] X[5] 2 3 + 17211630765487082247149408699048 X[2] X[4] 2 2 - 26789996015818372263330520019496 X[2] X[4] X[5] 2 2 + 13659237196782042217740944201316 X[2] X[4] X[5] 2 3 - 647804889685913518650889484730 X[2] X[5] 4 + 9540178380938185006260088289849 X[2] X[3] 3 + 41596173550585930875569967708088 X[2] X[3] X[4] 3 - 34091047509268100522472925931268 X[2] X[3] X[5] 2 2 + 56757184277544828042803967883008 X[2] X[3] X[4] 2 - 83082417004895768759594869265772 X[2] X[3] X[4] X[5] 2 2 + 11118097481843836746767489820030 X[2] X[3] X[5] 3 + 21678104396671932934673969709328 X[2] X[3] X[4] 2 - 39919991888391123337126446960288 X[2] X[3] X[4] X[5] 2 + 17682671606900184928795802775120 X[2] X[3] X[4] X[5] 3 - 736828143320784701286442165428 X[2] X[3] X[5] 4 - 2391477250252599220395997338016 X[2] X[4] 3 + 592705843038409507224874322976 X[2] X[4] X[5] 2 2 + 3642145655114326555203599061936 X[2] X[4] X[5] 3 - 580591488122787931373714306940 X[2] X[4] X[5] 4 - 9992466923334492859292608479 X[2] X[5] 5 + 1667477253584374997367727582795 X[3] 4 + 8546426017123703834071920838288 X[3] X[4] 4 - 4774177736111617988837780883123 X[3] X[5] 3 2 + 14013946047586863789708502481344 X[3] X[4] 3 - 16849082781109760500088264689668 X[3] X[4] X[5] 3 2 + 1976108624540514956383013287854 X[3] X[5] 2 3 + 6298416079975539739418464848104 X[3] X[4] 2 2 - 14660430130434350334219204094344 X[3] X[4] X[5] 2 2 + 5347944269810762340182717135268 X[3] X[4] X[5] 2 3 - 183127723948401117657077210958 X[3] X[5] 4 - 1694532572912121462577743892672 X[3] X[4] 3 + 360233201586660514772690038464 X[3] X[4] X[5] 2 2 + 3067859593990968459308736850944 X[3] X[4] X[5] 3 - 441873624589499639672198203068 X[3] X[4] X[5] 4 - 10085326334476827675589064313 X[3] X[5] 5 + 63717774936361494114299127200 X[4] 4 + 26973958779768611598096794688 X[4] X[5] 3 2 - 143916463246605313320858483720 X[4] X[5] 2 3 - 13704857162555046566196814488 X[4] X[5] 4 + 4573385427685651108837315308 X[4] X[5] 5 + 119502280377158108738317857 X[5] + 645387456420410425443606904131168 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j -11 t + 16 t + 16 t - 18 t - 18 ) a[1, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 5 t + 19 t - 2 t - 9 t + 1 j = 0 infinity ----- 4 3 2 \ j -13 t + 13 t + 6 t - 11 t + 5 ) a[2, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 5 t + 19 t - 2 t - 9 t + 1 j = 0 infinity ----- 4 3 \ j 18 t + 20 t - 12 t - 19 ) a[3, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 5 t + 19 t - 2 t - 9 t + 1 j = 0 infinity ----- 4 3 2 \ j -6 t - 11 t - 2 t - 8 t - 17 ) a[4, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 5 t + 19 t - 2 t - 9 t + 1 j = 0 infinity ----- 4 3 2 \ j 20 t - 13 t + 4 t - 9 t - 12 ) a[5, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 5 t + 19 t - 2 t - 9 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 3 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 177978621576084943886148249087 X[1] 4 - 1307887783478213907799360973546 X[1] X[2] 4 - 1804036671522125475019632275618 X[1] X[3] 4 + 1534256805656957311277616991008 X[1] X[4] 4 - 3002067216468206717255397257012 X[1] X[5] 3 2 + 3812852049158605273749241040143 X[1] X[2] 3 + 10579113603887983015985619338570 X[1] X[2] X[3] 3 - 9093553863928942015023018319769 X[1] X[2] X[4] 3 + 17600624645285250105809367875507 X[1] X[2] X[5] 3 2 + 7123774276991709438165639809689 X[1] X[3] 3 - 11882169151252643717902462812830 X[1] X[3] X[4] 3 + 23775151192579846340980084260875 X[1] X[3] X[5] 3 2 + 4874085892876632437009792991489 X[1] X[4] 3 - 19900396865745136108912423871498 X[1] X[4] X[5] 3 2 + 19815726502029804809078783769901 X[1] X[5] 2 3 - 5508791393370468987718542378381 X[1] X[2] 2 2 - 22935813162057996613340588950536 X[1] X[2] X[3] 2 2 + 19835087853403519370616026231784 X[1] X[2] X[4] 2 2 - 38181096646063051733627971164912 X[1] X[2] X[5] 2 2 - 31204341182842155985652575453107 X[1] X[2] X[3] 2 + 52564044090176285980000873342794 X[1] X[2] X[3] X[4] 2 - 103998181020762556427215136627430 X[1] X[2] X[3] X[5] 2 2 - 21739328747561661268783420814352 X[1] X[2] X[4] 2 + 87779851762665310959928111661286 X[1] X[2] X[4] X[5] 2 2 - 86576054733926059089007310311026 X[1] X[2] X[5] 2 3 - 13686646765548446319107982435243 X[1] X[3] 2 2 + 33540312604000951153641634790964 X[1] X[3] X[4] 2 2 - 68721537191423553491591883432996 X[1] X[3] X[5] 2 2 - 26840642920972134036787143468537 X[1] X[3] X[4] 2 + 112652648825027855895591218344860 X[1] X[3] X[4] X[5] 2 2 - 114830719811659293113794890532284 X[1] X[3] X[5] 2 3 + 6948993963093593685756914402685 X[1] X[4] 2 2 - 45201234858925955528371312990356 X[1] X[4] X[5] 2 2 + 94375695012873865310444658754275 X[1] X[4] X[5] 2 3 - 63858295803788834013830793897411 X[1] X[5] 4 + 3930173818274690510667829994659 X[1] X[2] 3 + 21774285498849654930546871443856 X[1] X[2] X[3] 3 - 18861178143492153664110307572921 X[1] X[2] X[4] 3 + 36276262159455150730737847286812 X[1] X[2] X[5] 2 2 + 44645287229734658678782286750619 X[1] X[2] X[3] 2 - 75625031414918176693210207778721 X[1] X[2] X[3] X[4] 2 + 148759284506376747701393787348795 X[1] X[2] X[3] X[5] 2 2 + 31407408260501188466341808464551 X[1] X[2] X[4] 2 - 126088652815109139456456870980655 X[1] X[2] X[4] X[5] 2 2 + 123802489596196523211420204924729 X[1] X[2] X[5] 3 + 39753426935457116710152268631713 X[1] X[2] X[3] 2 - 98436787730802477169709495368746 X[1] X[2] X[3] X[4] 2 + 199129214905619783309352504909453 X[1] X[2] X[3] X[5] 2 + 79615166112071571745906392333486 X[1] X[2] X[3] X[4] - 329312826758919298096131364429005 X[1] X[2] X[3] X[4] X[5] 2 + 332029333187287441921820865260091 X[1] X[2] X[3] X[5] 3 - 20891205293880768726296138465016 X[1] X[2] X[4] 2 + 133292805671722264571445537664101 X[1] X[2] X[4] X[5] 2 - 274866359955106781162564624972637 X[1] X[2] X[4] X[5] 3 + 184280181034455807673471569842515 X[1] X[2] X[5] 4 + 12807974057674293778148998035751 X[1] X[3] 3 - 41044047047873255138370067722948 X[1] X[3] X[4] 3 + 86022446121790483985721855569743 X[1] X[3] X[5] 2 2 + 48252108797858715086390172531711 X[1] X[3] X[4] 2 - 207432428626871313920239270271979 X[1] X[3] X[4] X[5] 2 2 + 216193304553929807166877317017775 X[1] X[3] X[5] 3 - 24510385192596967184685108865479 X[1] X[3] X[4] 2 + 163072301404713741285184439846814 X[1] X[3] X[4] X[5] 2 - 348447643657871772990577298584056 X[1] X[3] X[4] X[5] 3 + 240977857379590277455307747616382 X[1] X[3] X[5] 4 + 4523255414687467404110553569037 X[1] X[4] 3 - 41552252074152686619599873381550 X[1] X[4] X[5] 2 2 + 137252221590141299414608547532171 X[1] X[4] X[5] 3 - 194552330101191090487086553752255 X[1] X[4] X[5] 4 + 100514907391733200448517055027693 X[1] X[5] 5 - 1101887894318456324626240804247 X[2] 4 - 7612310782385702788189356682297 X[2] X[3] 4 + 6583139005238981655046550680636 X[2] X[4] 4 - 12696517301350399562995406851189 X[2] X[5] 3 2 - 20852405223690821968083754234826 X[2] X[3] 3 + 35394601695013078222201683285940 X[2] X[3] X[4] 3 - 69521387867094485988001401177829 X[2] X[3] X[5] 3 2 - 14725356487343557313586893205597 X[2] X[4] 3 + 58991790414792837425770838897374 X[2] X[4] X[5] 3 2 - 57884933492069229406553277036389 X[2] X[5] 2 3 - 28103261933995252830030678451445 X[2] X[3] 2 2 + 70034421264743510031771173105115 X[2] X[3] X[4] 2 2 - 140682330649105657839255620041881 X[2] X[3] X[5] 2 2 - 57031436639266362121575905705817 X[2] X[3] X[4] 2 + 233871522856960449368066402628732 X[2] X[3] X[4] X[5] 2 2 - 234434912548445973513247218520596 X[2] X[3] X[5] 2 3 + 15105353975546929925304658934853 X[2] X[4] 2 2 - 95192762258766222280592754328152 X[2] X[4] X[5] 2 2 + 194867451011085149117702266301445 X[2] X[4] X[5] 2 3 - 130034992207718266860391530503789 X[2] X[5] 4 - 18470712621323007204703457744221 X[2] X[3] 3 + 59894101577029615853155159351621 X[2] X[3] X[4] 3 - 123668642524470331548701286550492 X[2] X[3] X[5] 2 2 - 71357126229577482962236994093193 X[2] X[3] X[4] 2 + 301250331386346674321207169694374 X[2] X[3] X[4] X[5] 2 2 - 309949812071033972710357980157947 X[2] X[3] X[5] 3 + 36848520082839286016117434430325 X[2] X[3] X[4] 2 - 239502458983473638326537862049657 X[2] X[3] X[4] X[5] 2 + 503855434324277693956758337362696 X[2] X[3] X[4] X[5] 3 - 344622794690321036326782333250372 X[2] X[3] X[5] 4 - 6935863966161995305527756162723 X[2] X[4] 3 + 61864780355802999994616220474927 X[2] X[4] X[5] 2 2 - 200325464109255069872952344634663 X[2] X[4] X[5] 3 + 280203240922589144313026870790640 X[2] X[4] X[5] 4 - 143414543914740654606161969040238 X[2] X[5] 5 - 4678465947812265241563317202995 X[3] 4 + 18425266757960610514980626973589 X[3] X[4] 4 - 39407179600759688350835887180882 X[3] X[5] 3 2 - 28390085310220502230146153435444 X[3] X[4] 3 + 124560386674289321966538434093740 X[3] X[4] X[5] 3 2 - 132442239895409398818812184436379 X[3] X[5] 2 3 + 21291474459733719007498443572571 X[3] X[4] 2 2 - 144412821319167653433082259746797 X[3] X[4] X[5] 2 2 + 314768290147778997655862777480463 X[3] X[4] X[5] 2 3 - 222009525769956995602508490473993 X[3] X[5] 4 - 7746322294013015156761417510431 X[3] X[4] 3 + 72472073136566543054605817762751 X[3] X[4] X[5] 2 2 - 243856470805443802467088931122443 X[3] X[4] X[5] 3 + 352388680253971436462516650265767 X[3] X[4] X[5] 4 - 185609526692916636734859725147605 X[3] X[5] 5 + 1092170424078420566809879750779 X[4] 4 - 13249329119987453871679344289041 X[4] X[5] 3 2 + 61341063006319576071154438235526 X[4] X[5] 2 3 - 136685547055394017045777311501987 X[4] X[5] 4 + 147453208038930347928921447999109 X[4] X[5] 5 - 61911921951560338665561914262287 X[5] + 70597459182774839137771992788277 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j -17 t + 20 t - 12 t + 13 ) a[1, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 2 t - 4 t - 15 t - 4 t + 1 j = 0 infinity ----- 4 3 2 \ j 15 t + 15 t - 15 t + 15 t - 2 ) a[2, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 2 t - 4 t - 15 t - 4 t + 1 j = 0 infinity ----- 4 3 2 \ j -5 t + 2 t - 19 t + 19 t - 11 ) a[3, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 2 t - 4 t - 15 t - 4 t + 1 j = 0 infinity ----- 4 3 2 \ j -16 t - 18 t - 10 t + 7 t + 13 ) a[4, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 2 t - 4 t - 15 t - 4 t + 1 j = 0 infinity ----- 4 3 2 \ j -17 t - 10 t + 16 t - 10 t + 18 ) a[5, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 2 t - 4 t - 15 t - 4 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 4 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 13836368090620477574828800 X[1] 4 + 83404422715119845098168000 X[1] X[2] 4 + 48664451570668903916884080 X[1] X[3] 4 - 173357581954194432478043040 X[1] X[4] 4 - 32549982475768106997025840 X[1] X[5] 3 2 + 284817182813741935210542720 X[1] X[2] 3 + 531103431425618996788123680 X[1] X[2] X[3] 3 - 287317176249913906034707520 X[1] X[2] X[4] 3 - 186714759082015063964914080 X[1] X[2] X[5] 3 2 + 5795010888214894279279552 X[1] X[3] 3 - 1080638732972255873406034272 X[1] X[3] X[4] 3 - 469668539195118317311147072 X[1] X[3] X[5] 3 2 - 148941456575409040013204672 X[1] X[4] 3 - 568520111270956460334129824 X[1] X[4] X[5] 3 2 - 49687035137755126712892032 X[1] X[5] 2 3 + 492677791532421891040509440 X[1] X[2] 2 2 + 1899915552824556095048822400 X[1] X[2] X[3] 2 2 + 693775830638266836544599680 X[1] X[2] X[4] 2 2 - 197565920497983948494142720 X[1] X[2] X[5] 2 2 + 1655347559486009145858096064 X[1] X[2] X[3] 2 - 115716390319799351464400544 X[1] X[2] X[3] X[4] 2 - 1557781137168997102315273824 X[1] X[2] X[3] X[5] 2 2 + 732722766525783094313182656 X[1] X[2] X[4] 2 - 1822423111838101769521855008 X[1] X[2] X[4] X[5] 2 2 - 323861814794545620918156384 X[1] X[2] X[5] 2 3 + 993837606505587543709912096 X[1] X[3] 2 2 + 5726804088314197279028270560 X[1] X[3] X[4] 2 2 - 209476548609193729534423088 X[1] X[3] X[5] 2 2 + 18806555146843416610483654208 X[1] X[3] X[4] 2 + 4413775577602378230197257184 X[1] X[3] X[4] X[5] 2 2 + 364303629406725199761357536 X[1] X[3] X[5] 2 3 + 15361451752638309140857992704 X[1] X[4] 2 2 + 8483676422735048022680055360 X[1] X[4] X[5] 2 2 + 1624209369036698260281788288 X[1] X[4] X[5] 2 3 + 151579807237504794897035952 X[1] X[5] 4 + 394681455954816759456972800 X[1] X[2] 3 + 2434552087505926949873367680 X[1] X[2] X[3] 3 + 1625231174158050246777345280 X[1] X[2] X[4] 3 + 50965619329653716530272640 X[1] X[2] X[5] 2 2 + 3794680959207407267052681120 X[1] X[2] X[3] 2 + 1827414991710416162106679680 X[1] X[2] X[3] X[4] 2 - 1771025364225779547935981120 X[1] X[2] X[3] X[5] 2 2 - 739682570376889951584220160 X[1] X[2] X[4] 2 - 2992223398859097475935291520 X[1] X[2] X[4] X[5] 2 2 - 591146709986776629231101920 X[1] X[2] X[5] 3 + 2682981686576402506574038440 X[1] X[2] X[3] 2 + 6265998188520305073559722032 X[1] X[2] X[3] X[4] 2 - 1623129321169477254297821272 X[1] X[2] X[3] X[5] 2 + 19604720671329572962037705408 X[1] X[2] X[3] X[4] + 4761352596331991228181944800 X[1] X[2] X[3] X[4] X[5] 2 + 327444243021350289056039832 X[1] X[2] X[3] X[5] 3 + 17748154644685347640565875328 X[1] X[2] X[4] 2 + 13931533646870608986822134208 X[1] X[2] X[4] X[5] 2 + 3492848693923743223713336112 X[1] X[2] X[4] X[5] 3 + 346715558395619021388359832 X[1] X[2] X[5] 4 + 2013909964500444208711080858 X[1] X[3] 3 + 13449767557202640070169984528 X[1] X[3] X[4] 3 + 2640790053634103595092720656 X[1] X[3] X[5] 2 2 + 34672946936696894248711915600 X[1] X[3] X[4] 2 + 21396562313539377622324611168 X[1] X[3] X[4] X[5] 2 2 + 2688928473088602110362224340 X[1] X[3] X[5] 3 + 19960885299895231330760510240 X[1] X[3] X[4] 2 + 36244311229504282315740448896 X[1] X[3] X[4] X[5] 2 + 10249914318798833772215900400 X[1] X[3] X[4] X[5] 3 + 878082325426750172291229184 X[1] X[3] X[5] 4 - 10876451121494695236702661664 X[1] X[4] 3 + 8887997043439192219878553888 X[1] X[4] X[5] 2 2 + 5951258186802654156274273136 X[1] X[4] X[5] 3 + 993834690567968080621825952 X[1] X[4] X[5] 4 + 59081307496143327776446338 X[1] X[5] 5 + 117239133505775380884638720 X[2] 4 + 1015185911047954797620414720 X[2] X[3] 4 + 815444162454887398006059520 X[2] X[4] 4 + 93804214050982079973216000 X[2] X[5] 3 2 + 2291608422805072607527062272 X[2] X[3] 3 + 1379539052848113520817571968 X[2] X[3] X[4] 3 - 560826301699975905644348032 X[2] X[3] X[5] 3 2 - 1181159888516587724123109632 X[2] X[4] 3 - 1529671417319029350542013824 X[2] X[4] X[5] 3 2 - 292200703429491073078065792 X[2] X[5] 2 3 - 78575465831260051419748560 X[2] X[3] 2 2 - 11981451028197276796155456064 X[2] X[3] X[4] 2 2 - 4273404674551690575877625520 X[2] X[3] X[5] 2 2 - 23877984440275474495294608576 X[2] X[3] X[4] 2 - 11475700252846294455201788416 X[2] X[3] X[4] X[5] 2 2 - 1427319197134105080297923760 X[2] X[3] X[5] 2 3 - 12463218498007215578087316096 X[2] X[4] 2 2 - 5507762342732582867462025792 X[2] X[4] X[5] 2 2 - 791209204956065996542304576 X[2] X[4] X[5] 2 3 - 17182493219200993445408080 X[2] X[5] 4 - 3339587476561196903141112032 X[2] X[3] 3 - 29337456464616413006081745000 X[2] X[3] X[4] 3 - 3047014506141076075043200216 X[2] X[3] X[5] 2 2 - 79470680329693312166795788784 X[2] X[3] X[4] 2 - 5641657864263955129588587448 X[2] X[3] X[4] X[5] 2 2 + 2201786229892118402948943976 X[2] X[3] X[5] 3 - 105993455802550073156535439680 X[2] X[3] X[4] 2 - 3809604745565174104670371488 X[2] X[3] X[4] X[5] 2 + 11628596598562358458631815304 X[2] X[3] X[4] X[5] 3 + 1738620451654982201790276632 X[2] X[3] X[5] 4 - 60306217213099578207493642432 X[2] X[4] 3 - 9498812980251939805789847232 X[2] X[4] X[5] 2 2 + 9296389379068593194915585616 X[2] X[4] X[5] 3 + 2997007978559479035785220440 X[2] X[4] X[5] 4 + 256576301091753451286970744 X[2] X[5] 5 - 5527027823731956967895611049 X[3] 4 - 62438820810619194939053657252 X[3] X[4] 4 - 7915417618041548101865097237 X[3] X[5] 3 2 - 282643121794376793707163191832 X[3] X[4] 3 - 77610686698914015406978778728 X[3] X[4] X[5] 3 2 - 5319483605849705549282367930 X[3] X[5] 2 3 - 644867622269483834469982229248 X[3] X[4] 2 2 - 295846025280809470485439435112 X[3] X[4] X[5] 2 2 - 47411156421550765214761904640 X[3] X[4] X[5] 2 3 - 2940040126328146381250248650 X[3] X[5] 4 - 722781705963405832820769224512 X[3] X[4] 3 - 487506818124163467715317690752 X[3] X[4] X[5] 2 2 - 130531674425978343466184890696 X[3] X[4] X[5] 3 - 17689655071086640302079975640 X[3] X[4] X[5] 4 - 1053152248152356750018625389 X[3] X[5] 5 - 313364515036443714845572180384 X[4] 4 - 285165123294028565713002935008 X[4] X[5] 3 2 - 109474338088742667759334936768 X[4] X[5] 2 3 - 23230288994731258040891311672 X[4] X[5] 4 - 2784194042283359267122778812 X[4] X[5] 5 - 146778106571103594015032849 X[5] + 73996305819679016287558245376 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j -5 t + 3 t + 19 t + 11 t - 15 ) a[1, j] t = ------------------------------------ / 5 4 3 2 ----- -t + 9 t - 10 t + 13 t + 8 t + 1 j = 0 infinity ----- 4 3 2 \ j -18 t + 20 t + 2 t - 10 t + 11 ) a[2, j] t = ------------------------------------ / 5 4 3 2 ----- -t + 9 t - 10 t + 13 t + 8 t + 1 j = 0 infinity ----- 4 3 2 \ j 15 t - 19 t - 16 t - 20 t - 4 ) a[3, j] t = ------------------------------------ / 5 4 3 2 ----- -t + 9 t - 10 t + 13 t + 8 t + 1 j = 0 infinity ----- 4 3 2 \ j -4 t + 10 t + 9 t + 11 t - 1 ) a[4, j] t = ------------------------------------ / 5 4 3 2 ----- -t + 9 t - 10 t + 13 t + 8 t + 1 j = 0 infinity ----- 4 3 2 \ j -9 t - 19 t + 18 t - 18 t + 10 ) a[5, j] t = ------------------------------------ / 5 4 3 2 ----- -t + 9 t - 10 t + 13 t + 8 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 5 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 346637230166458734872687927903 X[1] 4 - 1981106029359902884718144772263 X[1] X[2] 4 - 3006314659458647944082778128636 X[1] X[3] 4 + 5884597378564518626597283981538 X[1] X[4] 4 + 4448067561172710619486717067686 X[1] X[5] 3 2 - 9145660844992463561745591514598 X[1] X[2] 3 - 1231486923229909856276826695092 X[1] X[2] X[3] 3 + 12357420311920512430513794937440 X[1] X[2] X[4] 3 + 301773270100003203528640703040 X[1] X[2] X[5] 3 2 + 6334938682901415183657361374604 X[1] X[3] 3 - 19131854563127074334208716127220 X[1] X[3] X[4] 3 - 19271925495277461173093438444060 X[1] X[3] X[5] 3 2 + 11256228249069179627468528302076 X[1] X[4] 3 + 29561940677538272495509284191548 X[1] X[4] X[5] 3 2 + 14331879944325778990830174953600 X[1] X[5] 2 3 + 38949134949884198610913305142942 X[1] X[2] 2 2 + 93193291038410749750565850889156 X[1] X[2] X[3] 2 2 - 207267969301271851760212664198220 X[1] X[2] X[4] 2 2 - 123853772460112659164140440209108 X[1] X[2] X[5] 2 2 + 53563554283936565426573996324284 X[1] X[2] X[3] 2 - 269936338688180233957270198310892 X[1] X[2] X[3] X[4] 2 - 140436499910954928510124174688692 X[1] X[2] X[3] X[5] 2 2 + 324936091835106912535596065603332 X[1] X[2] X[4] 2 + 359644490748872292180828701598684 X[1] X[2] X[4] X[5] 2 2 + 94321563439968676533564163874560 X[1] X[2] X[5] 2 3 + 3063105470280556463530811749896 X[1] X[3] 2 2 - 44684363125346671291023399527280 X[1] X[3] X[4] 2 2 - 8890337762903957713582517886448 X[1] X[3] X[5] 2 2 + 146991355337348695133154633201816 X[1] X[3] X[4] 2 + 111164086975468042234210697055424 X[1] X[3] X[4] X[5] 2 2 + 9253512176678182584213610652168 X[1] X[3] X[5] 2 3 - 138871420871773054567817031491448 X[1] X[4] 2 2 - 194208724051065485024770048692352 X[1] X[4] X[5] 2 2 - 73598090828025059935911393551528 X[1] X[4] X[5] 2 3 - 4047027303386566267596724374296 X[1] X[5] 4 + 98918942467637818640652102345095 X[1] X[2] 3 + 161454666203075412897763949584212 X[1] X[2] X[3] 3 - 510190697666396242140442402410112 X[1] X[2] X[4] 3 - 251537721144557169641865858018000 X[1] X[2] X[5] 2 2 + 56905284720939316609039627434772 X[1] X[2] X[3] 2 - 516739718405484594131385930478028 X[1] X[2] X[3] X[4] 2 - 210963108207034478279937890201028 X[1] X[2] X[3] X[5] 2 2 + 915868053989152159182411725112276 X[1] X[2] X[4] 2 + 842143345191510846730598710124068 X[1] X[2] X[4] X[5] 2 2 + 178709321290817592146111399626880 X[1] X[2] X[5] 3 - 9449968021683749506082274791216 X[1] X[2] X[3] 2 - 56790849995227586179039250966832 X[1] X[2] X[3] X[4] 2 + 6054023260685724773819395721568 X[1] X[2] X[3] X[5] 2 + 471037823354407769850649030577376 X[1] X[2] X[3] X[4] + 307361402081704078038787591502896 X[1] X[2] X[3] X[4] X[5] 2 + 26959447116613767261626136542800 X[1] X[2] X[3] X[5] 3 - 664907676817314248939017735112688 X[1] X[2] X[4] 2 - 826713029933501737593403759885344 X[1] X[2] X[4] X[5] 2 - 294178090815887008294167510383600 X[1] X[2] X[4] X[5] 3 - 23477049765262053877098317609600 X[1] X[2] X[5] 4 - 2044717667463037476261630570800 X[1] X[3] 3 + 12441958351846997179005463387968 X[1] X[3] X[4] 3 + 4212349150816772985728332403760 X[1] X[3] X[5] 2 2 + 7556673428466526826226453210400 X[1] X[3] X[4] 2 - 8048086434323666274484952977248 X[1] X[3] X[4] X[5] 2 2 + 375730871231297943698153362928 X[1] X[3] X[5] 3 - 130870126865996567093799624650160 X[1] X[3] X[4] 2 - 113091693099871434388262349084480 X[1] X[3] X[4] X[5] 2 - 24286688949131259158515448688016 X[1] X[3] X[4] X[5] 3 - 3073013185509656477616660357616 X[1] X[3] X[5] 4 + 166753725565385545207441446492352 X[1] X[4] 3 + 248894373230339934046412723856128 X[1] X[4] X[5] 2 2 + 114802272215209853226395096335968 X[1] X[4] X[5] 3 + 15017871959479479706038105205456 X[1] X[4] X[5] 4 - 141918383647930185783640654480 X[1] X[5] 5 + 48480082198372090466732561099945 X[2] 4 + 58026477248722388921111934314840 X[2] X[3] 4 - 281831850396459896914517825154390 X[2] X[4] 4 - 127568817896085143008363578667490 X[2] X[5] 3 2 + 100157027879469623356624987908 X[2] X[3] 3 - 221025621938124902088064808416116 X[2] X[3] X[4] 3 - 78519357047257139496363727371660 X[2] X[3] X[5] 3 2 + 618175979030033736621556895734188 X[2] X[4] 3 + 526618770929134925264726496701380 X[2] X[4] X[5] 3 2 + 103658410062504636120057599212960 X[2] X[5] 2 3 - 14523140586132895410478678796024 X[2] X[3] 2 2 + 23826523619082084080957339772736 X[2] X[3] X[4] 2 2 + 20784333526315318132330980932848 X[2] X[3] X[5] 2 2 + 286299731389967633112400276313160 X[2] X[3] X[4] 2 + 174528546737520388380313928560784 X[2] X[3] X[4] X[5] 2 2 + 18084345704483033729755041767560 X[2] X[3] X[5] 2 3 - 633740390672284191743865057849592 X[2] X[4] 2 2 - 749369087673481771944335330036032 X[2] X[4] X[5] 2 2 - 262562185251053316065392307594120 X[2] X[4] X[5] 2 3 - 24228686327469216352516635497640 X[2] X[5] 4 - 1555533888575563594051871683760 X[2] X[3] 3 + 21181370693561879076417479502432 X[2] X[3] X[4] 3 + 4034072832844127925918796782832 X[2] X[3] X[5] 2 2 - 30416404357275033172896210178080 X[2] X[3] X[4] 2 - 27001801294927426621448930689728 X[2] X[3] X[4] X[5] 2 2 - 353561837597001540850066425520 X[2] X[3] X[5] 3 - 153940652742299249045993058920656 X[2] X[3] X[4] 2 - 127697681451060016867300646679520 X[2] X[3] X[4] X[5] 2 - 28987861811653072308393400440080 X[2] X[3] X[4] X[5] 3 - 3292275479784486470864159926960 X[2] X[3] X[5] 4 + 302456656836002123335467998913184 X[2] X[4] 3 + 435948176102221896883286373458976 X[2] X[4] X[5] 2 2 + 200791585092826458779938283581760 X[2] X[4] X[5] 3 + 29114516186307440050847356906960 X[2] X[4] X[5] 4 + 82659935872157964234955360880 X[2] X[5] 5 + 136331594644270959051706116064 X[3] 4 + 234857276009134819117667678976 X[3] X[4] 4 - 228771684582143091205228217824 X[3] X[5] 3 2 - 6473299105789821570688773215648 X[3] X[4] 3 - 1874641900450345606249328308864 X[3] X[4] X[5] 3 2 - 150605253649383948782861895968 X[3] X[5] 2 3 + 9715130292289283913526492052864 X[3] X[4] 2 2 + 9586639934678296371058647426272 X[3] X[4] X[5] 2 2 + 1197771987225580241958200921824 X[3] X[4] X[5] 2 3 + 219683600679482781491198065376 X[3] X[5] 4 + 28536867612188101367032881584512 X[3] X[4] 3 + 28241282808048533123928279772064 X[3] X[4] X[5] 2 2 + 8864896411451452451873300543136 X[3] X[4] X[5] 3 + 1575255813037479361641661488288 X[3] X[4] X[5] 4 + 89935047726313878595895999040 X[3] X[5] 5 - 53671303481195038932789798173728 X[4] 4 - 87936914484237558076048369360128 X[4] X[5] 3 2 - 47539212437138329391581191199872 X[4] X[5] 2 3 - 8664856131857418254145438367584 X[4] X[5] 4 - 116832368186977786725003285760 X[4] X[5] 5 + 5273613918211507896201369440 X[5] + 5061063302733364595339913156773024 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j t - 12 t + 2 t - 7 t - 13 ) a[1, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 3 t + 15 t + 5 t - 6 t + 1 j = 0 infinity ----- 4 3 2 \ j 13 t - 18 t - 12 t + 13 t + 11 ) a[2, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 3 t + 15 t + 5 t - 6 t + 1 j = 0 infinity ----- 4 3 2 \ j 16 t + 3 t - 12 t - 18 t - 16 ) a[3, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 3 t + 15 t + 5 t - 6 t + 1 j = 0 infinity ----- 4 3 2 \ j 20 t - 18 t + 8 t - 5 t + 9 ) a[4, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 3 t + 15 t + 5 t - 6 t + 1 j = 0 infinity ----- 4 3 2 \ j -11 t + 20 t - 14 t + 19 t - 18 ) a[5, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 3 t + 15 t + 5 t - 6 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 6 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 290991303800208777902693437277 X[1] 4 - 140489529174474035873780514322 X[1] X[2] 4 - 747702704906452000167709985034 X[1] X[3] 4 - 771949645264262426207267088721 X[1] X[4] 4 + 1406453084920294891642596220371 X[1] X[5] 3 2 - 3159783983814656310624760060 X[1] X[2] 3 + 289881710489654718760432690512 X[1] X[2] X[3] 3 + 289207615319966069562883980232 X[1] X[2] X[4] 3 - 565656881508192066581574616176 X[1] X[2] X[5] 3 2 + 754007366069425229018320076796 X[1] X[3] 3 + 1607528272709483176082785636704 X[1] X[3] X[4] 3 - 2929112293963606303332272524992 X[1] X[3] X[5] 3 2 + 808578287950621829966087736182 X[1] X[4] 3 - 2956160598264309982786353129780 X[1] X[4] X[5] 3 2 + 2692348786880905285805139650322 X[1] X[5] 2 3 + 151212549359379081325143736 X[1] X[2] 2 2 + 11045559846918647811998954088 X[1] X[2] X[3] 2 2 + 973177872764827533015998244 X[1] X[2] X[4] 2 2 - 5934403384529804325886379724 X[1] X[2] X[5] 2 2 - 218715049326397566198782641752 X[1] X[2] X[3] 2 - 454832703107709681064720697856 X[1] X[2] X[3] X[4] 2 + 896111348745616580211253391952 X[1] X[2] X[3] X[5] 2 2 - 219947779078889006497297713972 X[1] X[2] X[4] 2 + 856822241255090402281892280768 X[1] X[2] X[4] X[5] 2 2 - 842893394773841854005232260900 X[1] X[2] X[5] 2 3 - 373286751275237920365723872328 X[1] X[3] 2 2 - 1229940949406360008982170795668 X[1] X[3] X[4] 2 2 + 2243542532290077492752335520076 X[1] X[3] X[5] 2 2 - 1282462956090850512137943154428 X[1] X[3] X[4] 2 + 4683498067194480179501255165160 X[1] X[3] X[4] X[5] 2 2 - 4261763118709121384823046436196 X[1] X[3] X[5] 2 3 - 416590402167360723440296903598 X[1] X[4] 2 2 + 2294905055245849731195712721502 X[1] X[4] X[5] 2 2 - 4200157927676284450274737271070 X[1] X[4] X[5] 2 3 + 2549206892961415399695389161974 X[1] X[5] 4 + 199003790689228634400163648 X[1] X[2] 3 + 167536935034776226094290368 X[1] X[2] X[3] 3 - 96202769612103654347321456 X[1] X[2] X[4] 3 + 1421600044774753136461909392 X[1] X[2] X[5] 2 2 - 9246070117408809116742994944 X[1] X[2] X[3] 2 - 6525931666028112873860742288 X[1] X[2] X[3] X[4] 2 + 18996260060468928985534437552 X[1] X[2] X[3] X[5] 2 2 + 1377134023430780941674228252 X[1] X[2] X[4] 2 - 1320752186529055125870128328 X[1] X[2] X[4] X[5] 2 2 - 91504193064059177194432500 X[1] X[2] X[5] 3 + 71506758468466830457964259648 X[1] X[2] X[3] 2 + 231944741408619705433394086704 X[1] X[2] X[3] X[4] 2 - 461772821552250889669434598224 X[1] X[2] X[3] X[5] 2 + 235496236209948904601494368144 X[1] X[2] X[3] X[4] - 920518212149695828345633475328 X[1] X[2] X[3] X[4] X[5] 2 + 912547664158687696012123849968 X[1] X[2] X[3] X[5] 3 + 72760045681787862892298445400 X[1] X[2] X[4] 2 - 424761027969488904638248166208 X[1] X[2] X[4] X[5] 2 + 833653496122762136151148972776 X[1] X[2] X[4] X[5] 3 - 548196865490274733611301164192 X[1] X[2] X[5] 4 + 90892997629158854526668178624 X[1] X[3] 3 + 409882813817198163606827518032 X[1] X[3] X[4] 3 - 749226171685228580523104841456 X[1] X[3] X[5] 2 2 + 663249472542802484497901482308 X[1] X[3] X[4] 2 - 2422586768134147427219700063960 X[1] X[3] X[4] X[5] 2 2 + 2204722702246265559860134502772 X[1] X[3] X[5] 3 + 448803393143022381644256768864 X[1] X[3] X[4] 2 - 2465335508070732873645422590272 X[1] X[3] X[4] X[5] 2 + 4502136631396249858589584832592 X[1] X[3] X[4] X[5] 3 - 2726487341045163279906682846032 X[1] X[3] X[5] 4 + 105045991714326465808230993805 X[1] X[4] 3 - 776602397271092890693499015724 X[1] X[4] X[5] 2 2 + 2145408344332953517452165193890 X[1] X[4] X[5] 3 - 2621187372677474602734480610020 X[1] X[4] X[5] 4 + 1193204584643879386104472774593 X[1] X[5] 5 4 - 6253821006319728414183776 X[2] - 155179759558095751225455456 X[2] X[3] 4 - 59821376418773032192546336 X[2] X[4] 4 + 102907709633805148060088256 X[2] X[5] 3 2 - 225746236605102691342341312 X[2] X[3] 3 - 19019522916001213890706752 X[2] X[3] X[4] 3 - 820780621005225598826746176 X[2] X[3] X[5] 3 2 + 12876868118881136912213272 X[2] X[4] 3 - 440284749076634777740167696 X[2] X[4] X[5] 3 2 + 860702730587905973327377848 X[2] X[5] 2 3 + 2171481164064705021565684032 X[2] X[3] 2 2 + 3451895549114736133993953600 X[2] X[3] X[4] 2 2 - 9135469627425046346222033280 X[2] X[3] X[5] 2 2 + 421877912386833516343991112 X[2] X[3] X[4] 2 - 4616476940113376603679238896 X[2] X[3] X[4] X[5] 2 2 + 5670908230689859031193525480 X[2] X[3] X[5] 2 3 - 511615370995087885310770628 X[2] X[4] 2 2 + 2080221820489787780082920868 X[2] X[4] X[5] 2 2 - 2998426427667203698006626780 X[2] X[4] X[5] 2 3 + 1952387382750553611007598268 X[2] X[5] 4 - 8575974482949891990388414944 X[2] X[3] 3 - 38290757391926702102991358656 X[2] X[3] X[4] 3 + 77261487346127481087310479936 X[2] X[3] X[5] 2 2 - 61169988737342940659034444216 X[2] X[3] X[4] 2 + 240760099762732093956363700560 X[2] X[3] X[4] X[5] 2 2 - 241588282706601641590063997016 X[2] X[3] X[5] 3 - 40030141860395484396194135328 X[2] X[3] X[4] 2 + 233424715238375981114534174640 X[2] X[3] X[4] X[5] 2 - 459500778789883677246128393088 X[2] X[3] X[4] X[5] 3 + 303970266432742607800648694640 X[2] X[3] X[5] 4 - 8745435197427456528682546042 X[2] X[4] 3 + 68340634570363518567831044016 X[2] X[4] X[5] 2 2 - 201401883601230328619491741932 X[2] X[4] X[5] 3 + 265034381097107031638000243472 X[2] X[4] X[5] 4 - 130994222936036168076179868090 X[2] X[5] 5 - 8726745385987218087235971552 X[3] 4 - 50295343805417475080566735392 X[3] X[4] 4 + 92201355994994496170117931456 X[3] X[5] 3 2 - 111793482843569364154436334504 X[3] X[4] 3 + 408945865005126425976768927792 X[3] X[4] X[5] 3 2 - 372606504929663933189395585224 X[3] X[5] 2 3 - 118024077625325124518457902700 X[3] X[4] 2 2 + 647553512742489141874469147772 X[3] X[4] X[5] 2 2 - 1181573728499381982241496818068 X[3] X[4] X[5] 2 3 + 714656130167331268861136571972 X[3] X[5] 4 - 57915407956323192684176179482 X[3] X[4] 3 + 425878615445285635127381112696 X[3] X[4] X[5] 2 2 - 1171535006042622517919920177572 X[3] X[4] X[5] 3 + 1426070579838102956718635036904 X[3] X[4] X[5] 4 - 646774311536916932059312711554 X[3] X[5] 5 - 10291442959804858369954215809 X[4] 4 + 96046302429616772911651639695 X[4] X[5] 3 2 - 356919789646636081347905247942 X[4] X[5] 2 3 + 659694033144696134527318814922 X[4] X[5] 4 - 605756656394076489684143756409 X[4] X[5] 5 + 220797174567974675505124888071 X[5] + 35725051535537900984015821824 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j 16 t - 12 t - 9 t + 20 t - 12 ) a[1, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 2 t - 20 t + 9 t + 12 t + 1 j = 0 infinity ----- 4 3 2 \ j 10 t - 9 t + 16 t + 20 t + 3 ) a[2, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 2 t - 20 t + 9 t + 12 t + 1 j = 0 infinity ----- 4 3 2 \ j 12 t - 19 t + 7 t + 7 t - 13 ) a[3, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 2 t - 20 t + 9 t + 12 t + 1 j = 0 infinity ----- 4 3 2 \ j -12 t - 6 t - 7 t + 9 t - 12 ) a[4, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 2 t - 20 t + 9 t + 12 t + 1 j = 0 infinity ----- 4 3 2 \ j -16 t - 6 t + 8 t - 7 t ) a[5, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 2 t - 20 t + 9 t + 12 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 7 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 4514456034857395824715008 X[1] 4 - 4060451461458030358125888 X[1] X[2] 4 - 3528434524921617966902208 X[1] X[3] 4 - 23143065783341673193781184 X[1] X[4] 4 + 2915396718466230227717568 X[1] X[5] 3 2 - 29774942849023021455704928 X[1] X[2] 3 - 66665961789008631663787776 X[1] X[2] X[3] 3 + 259821825929457459157208448 X[1] X[2] X[4] 3 - 314179452359001076081656960 X[1] X[2] X[5] 3 2 - 43635584414372654577656736 X[1] X[3] 3 + 310048764585222376878698304 X[1] X[3] X[4] 3 - 367113016737419541948369216 X[1] X[3] X[5] 3 2 - 450346456440741939614206560 X[1] X[4] 3 + 1239578921022830648306631360 X[1] X[4] X[5] 3 2 - 803170587854520340947993888 X[1] X[5] 2 3 + 1021731244436998671771276 X[1] X[2] 2 2 - 31806045734502450516986196 X[1] X[2] X[3] 2 2 + 189980525590669236952669212 X[1] X[2] X[4] 2 2 - 63550321899294808761056460 X[1] X[2] X[5] 2 2 - 91008517356013584075381756 X[1] X[2] X[3] 2 + 759745677269253411745213032 X[1] X[2] X[3] X[4] 2 - 546980973856939139184811080 X[1] X[2] X[3] X[5] 2 2 - 1660648672106217537799684428 X[1] X[2] X[4] 2 + 2721403832569840683706766904 X[1] X[2] X[4] X[5] 2 2 - 768467605890080374166078460 X[1] X[2] X[5] 2 3 - 59095718305034129588978748 X[1] X[3] 2 2 + 695237067138790927946346876 X[1] X[3] X[4] 2 2 - 615845994823200379138448364 X[1] X[3] X[5] 2 2 - 2713797671850209416603150692 X[1] X[3] X[4] 2 + 5010428680859150364309136488 X[1] X[3] X[4] X[5] 2 2 - 2023812780670050810154994484 X[1] X[3] X[5] 2 3 + 3395646748984395178634111460 X[1] X[4] 2 2 - 10071112942594123587219362652 X[1] X[4] X[5] 2 2 + 8983648770420894297332495580 X[1] X[4] X[5] 2 3 - 2125515913207479625255378596 X[1] X[5] 4 + 26137499941369634886781572 X[1] X[2] 3 + 74936743173341513555602656 X[1] X[2] X[3] 3 - 337405809321629941431592896 X[1] X[2] X[4] 3 + 406647294652738015812109428 X[1] X[2] X[5] 2 2 + 37008387405370048546218888 X[1] X[2] X[3] 2 - 470118802572453882448509840 X[1] X[2] X[3] X[4] 2 + 681293719735800967128492036 X[1] X[2] X[3] X[5] 2 2 + 1092225473691723852478825248 X[1] X[2] X[4] 2 - 3403334284570798814615803164 X[1] X[2] X[4] X[5] 2 2 + 2187067557124565781883498836 X[1] X[2] X[5] 3 - 51664357525809548662885824 X[1] X[2] X[3] 2 + 395964767596437068253953760 X[1] X[2] X[3] X[4] 2 - 185669918199930537101512548 X[1] X[2] X[3] X[5] 2 - 953832161096943480282915648 X[1] X[2] X[3] X[4] - 439346337391172242472401512 X[1] X[2] X[3] X[4] X[5] 2 + 1223895053128438440422136648 X[1] X[2] X[3] X[5] 3 + 914954996484887731014501696 X[1] X[2] X[4] 2 + 2782800901669436191459590156 X[1] X[2] X[4] X[5] 2 - 8564792217689441160788364504 X[1] X[2] X[4] X[5] 3 + 4419318351557322979087712604 X[1] X[2] X[5] 4 - 40113540019803970558654764 X[1] X[3] 3 + 551351841767566066602216048 X[1] X[3] X[4] 3 - 488875447432479177725565972 X[1] X[3] X[5] 2 2 - 2879108737774847258137415712 X[1] X[3] X[4] 2 + 4692587514982311041328571524 X[1] X[3] X[4] X[5] 2 2 - 1827577032579263551827976668 X[1] X[3] X[5] 3 + 6812182166166169223771317728 X[1] X[3] X[4] 2 - 15410114845585097353942757724 X[1] X[3] X[4] X[5] 2 + 9766659534361949695531413384 X[1] X[3] X[4] X[5] 3 - 1570298332856723865890259420 X[1] X[3] X[5] 4 - 6157413598886713347837983268 X[1] X[4] 3 + 17830409199176667060022022700 X[1] X[4] X[5] 2 2 - 14034355840762308029219418756 X[1] X[4] X[5] 3 + 133485160925684031914597892 X[1] X[4] X[5] 4 + 1929980236685460628947663864 X[1] X[5] 5 + 34091853321801959987439407 X[2] 4 + 169782439355321442504060817 X[2] X[3] 4 - 614588646675739265440039931 X[2] X[4] 4 + 743782872539030663295222929 X[2] X[5] 3 2 + 319996062871779767670404342 X[2] X[3] 3 - 2336232173833452892511420180 X[2] X[3] X[4] 3 + 2896394066072160578158886732 X[2] X[3] X[5] 3 2 + 4326285634685926497353502659 X[2] X[4] 3 - 10512608107984981710841184698 X[2] X[4] X[5] 3 2 + 6379990411032989950743609074 X[2] X[5] 2 3 + 276007162174156649384464226 X[2] X[3] 2 2 - 3056257752983907404705625810 X[2] X[3] X[4] 2 2 + 3940852653963982762776757830 X[2] X[3] X[5] 2 2 + 11438642007332175950641012299 X[2] X[3] X[4] 2 - 28855205788611220179586770522 X[2] X[3] X[4] X[5] 2 2 + 18070498465647239274848512386 X[2] X[3] X[5] 2 3 - 14411443484462535949934636365 X[2] X[4] 2 2 + 53709458074070467039264089723 X[2] X[4] X[5] 2 2 - 65867620929952644967897370784 X[2] X[4] X[5] 2 3 + 26822956744125893906473217494 X[2] X[5] 4 + 100221998233470491523519019 X[2] X[3] 3 - 1507893350289378843951341876 X[2] X[3] X[4] 3 + 2101965544912033913290353356 X[2] X[3] X[5] 2 2 + 8600873149547524927808620305 X[2] X[3] X[4] 2 - 23396819247460376055539772222 X[2] X[3] X[4] X[5] 2 2 + 15514931810288983243564903830 X[2] X[3] X[5] 3 - 22035721716204638624551622030 X[2] X[3] X[4] 2 + 88174951888504022928717394002 X[2] X[3] X[4] X[5] 2 - 114202620303917443857673316880 X[2] X[3] X[4] X[5] 3 + 48566442129533079568781802116 X[2] X[3] X[5] 4 + 21243685844034585899459138434 X[2] X[4] 3 - 111923332646004844355979343804 X[2] X[4] X[5] 2 2 + 214003228100560031948582075433 X[2] X[4] X[5] 3 - 178042688192977123424093557342 X[2] X[4] X[5] 4 + 55038037402156582616690937007 X[2] X[5] 5 4 + 8547670938987685752247997 X[3] - 171246692696506806758058731 X[3] X[4] 4 + 310615679445867636254783249 X[3] X[5] 3 2 + 1370784437252376866653812257 X[3] X[4] 3 - 4765421052911708382969422254 X[3] X[4] X[5] 3 2 + 3653025291506439471850064630 X[3] X[5] 2 3 - 5496747772327340287794621301 X[3] X[4] 2 2 + 27643305580300247457606273795 X[3] X[4] X[5] 2 2 - 41110184825204770498117435392 X[3] X[4] X[5] 2 3 + 19183193739784837491006890134 X[3] X[5] 4 + 11046460813980773530385199382 X[3] X[4] 3 - 71903000742952595864235099172 X[3] X[4] X[5] 2 2 + 156554067180193411760594049147 X[3] X[4] X[5] 3 - 142335875868438022919671401514 X[3] X[4] X[5] 4 + 46938658706127261912472512061 X[3] X[5] 5 - 8861831656775745571821011308 X[4] 4 + 70368828177407760258509616986 X[4] X[5] 3 2 - 200771390588944139413359806107 X[4] X[5] 2 3 + 268943819780993962934402654209 X[4] X[5] 4 - 173302337288028901116850201469 X[4] X[5] 5 + 43789636231328719197572069737 X[5] + 573885202619564271384512256 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j 13 t + 3 t + 7 t - 17 t + 12 ) a[1, j] t = ----------------------------------- / 5 4 3 2 ----- -t + t - 18 t - 16 t + 16 t + 1 j = 0 infinity ----- 4 3 2 \ j 12 t - 16 t + t - 17 ) a[2, j] t = ----------------------------------- / 5 4 3 2 ----- -t + t - 18 t - 16 t + 16 t + 1 j = 0 infinity ----- 4 3 2 \ j 16 t + 13 t + 19 t + 13 t - 8 ) a[3, j] t = ----------------------------------- / 5 4 3 2 ----- -t + t - 18 t - 16 t + 16 t + 1 j = 0 infinity ----- 4 3 2 \ j 8 t + 14 t + 8 t - 20 t + 8 ) a[4, j] t = ----------------------------------- / 5 4 3 2 ----- -t + t - 18 t - 16 t + 16 t + 1 j = 0 infinity ----- 3 2 \ j 11 t - 2 t - 19 t + 13 ) a[5, j] t = ----------------------------------- / 5 4 3 2 ----- -t + t - 18 t - 16 t + 16 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 8 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = -76698468078065513247592082432 X[1] 4 + 199204510698568659939151539456 X[1] X[2] 4 - 354740699128719287700042395904 X[1] X[3] 4 - 22341116405584284981061946624 X[1] X[4] 4 + 316671937934963430509213218560 X[1] X[5] 3 2 - 214938259393502628690208448000 X[1] X[2] 3 + 722376688253171445604623746560 X[1] X[2] X[3] 3 + 35961252319738339260155319552 X[1] X[2] X[4] 3 - 667969766844574735036669317632 X[1] X[2] X[5] 3 2 - 641633072545985248493271985152 X[1] X[3] 3 - 74816120187514699143966351104 X[1] X[3] X[4] 3 + 1150366338167568524694109727744 X[1] X[3] X[5] 3 2 - 1739811773632064850055367424 X[1] X[4] 3 + 63847815862224809465184172288 X[1] X[4] X[5] 3 2 - 523385231701810948398165885952 X[1] X[5] 2 3 + 117055975631508448079766459504 X[1] X[2] 2 2 - 568898490861865792672902020176 X[1] X[2] X[3] 2 2 - 20636771467862473749205476432 X[1] X[2] X[4] 2 2 + 540500729962672224398147355824 X[1] X[2] X[5] 2 2 + 963359266483862171297535679824 X[1] X[2] X[3] 2 + 91245602081915711733445098912 X[1] X[2] X[3] X[4] 2 - 1778452005422199279267106137184 X[1] X[2] X[3] X[5] 2 2 + 2392031094322078293995307088 X[1] X[2] X[4] 2 - 75975698425229953611158976608 X[1] X[2] X[4] X[5] 2 2 + 833851084682219452348020947280 X[1] X[2] X[5] 2 3 - 565825966518868625607479274864 X[1] X[3] 2 2 - 89631344983551308307088529232 X[1] X[3] X[4] 2 2 + 1532084414032814285717033775536 X[1] X[3] X[5] 2 2 - 2734514885036461839327181392 X[1] X[3] X[4] 2 + 157846797299604562815568852064 X[1] X[3] X[4] X[5] 2 2 - 1397269465593555298940303373136 X[1] X[3] X[5] 2 3 + 207481342531939963546579856 X[1] X[4] 2 2 + 3531784206914111073813798576 X[1] X[4] X[5] 2 2 - 67340778279528033543852070736 X[1] X[4] X[5] 2 3 + 430902663033555724468523063696 X[1] X[5] 4 - 31603295290849931933581563372 X[1] X[2] 3 + 201304265687198347560019124128 X[1] X[2] X[3] 3 + 4963683231704628784372077584 X[1] X[2] X[4] 3 - 194739103418114791588107769072 X[1] X[2] X[5] 2 2 - 494111587809805331325884172760 X[1] X[2] X[3] 2 - 35514361905644672109317723712 X[1] X[2] X[3] X[4] 2 + 934562335714744273762685556096 X[1] X[2] X[3] X[5] 2 2 - 1073272470677347639430198440 X[1] X[2] X[4] 2 + 28632882394609586326678075248 X[1] X[2] X[4] X[5] 2 2 - 448147041181640270790727789736 X[1] X[2] X[5] 3 + 558499603281726451629564392576 X[1] X[2] X[3] 2 + 73732122952294074286712229712 X[1] X[2] X[3] X[4] 2 - 1548675393379565778445648574512 X[1] X[2] X[3] X[5] 2 + 2696800212348869159755105952 X[1] X[2] X[3] X[4] - 126931245974811747530796614336 X[1] X[2] X[3] X[4] X[5] 2 + 1448762327312240759352452059936 X[1] X[2] X[3] X[5] 3 - 166039136133245396976713776 X[1] X[2] X[4] 2 - 3301621172233058131826819088 X[1] X[2] X[4] X[5] 2 + 52668617448050081310788126704 X[1] X[2] X[4] X[5] 3 - 458272596588143988761547772720 X[1] X[2] X[5] 4 - 242909904027234324968201881692 X[1] X[3] 3 - 45609158730761389477114841376 X[1] X[3] X[4] 3 + 884526982242479613831786011552 X[1] X[3] X[5] 2 2 - 1111910357238609226323348088 X[1] X[3] X[4] 2 + 123923733662803831889115720016 X[1] X[3] X[4] X[5] 2 2 - 1216158337714468248577431865336 X[1] X[3] X[5] 3 + 297969349223258831907196160 X[1] X[3] X[4] 2 + 3507820419383310685476092992 X[1] X[3] X[4] X[5] 2 - 109258157128023211047876866304 X[1] X[3] X[4] X[5] 3 + 750753188371313277199353330368 X[1] X[3] X[5] 4 - 1688803848719842129352012 X[1] X[4] 3 - 293983305854464172179771760 X[1] X[4] X[5] 2 2 - 2348497074040691068385565512 X[1] X[4] X[5] 3 + 31053426616061581875710637968 X[1] X[4] X[5] 4 - 176128134751100218234276459852 X[1] X[5] 5 + 3354108145034582626373174219 X[2] 4 - 26619143320846696967576491783 X[2] X[3] 4 - 412745976962063866934036455 X[2] X[4] 4 + 26028077387261212749112594145 X[2] X[5] 3 2 + 85381206683363349773263509678 X[2] X[3] 3 + 4396606946570788675298296524 X[2] X[3] X[4] 3 - 164394794639840445018537051556 X[2] X[3] X[5] 3 2 + 157836238535935346676730126 X[2] X[4] 3 - 3383192042170459453633830708 X[2] X[4] X[5] 3 2 + 80024077612817483197952837118 X[2] X[5] 2 3 - 140469586993313679521373147278 X[2] X[3] 2 2 - 14599517905882017522315710010 X[2] X[3] X[4] 2 2 + 397727017544644840417608477734 X[2] X[3] X[5] 2 2 - 635370123362700309243682074 X[2] X[3] X[4] 2 + 24452288641967273962000038316 X[2] X[3] X[4] X[5] 2 2 - 379761121611679209641878594746 X[2] X[3] X[5] 2 3 + 33186422928961041787889170 X[2] X[4] 2 2 + 750537285750658655512797558 X[2] X[4] X[5] 2 2 - 9780085841473376453879668554 X[2] X[4] X[5] 2 3 + 122390374776325281897559313554 X[2] X[5] 4 + 118545753751197997931258400311 X[2] X[3] 3 + 18989666938331497719485430572 X[2] X[3] X[4] 3 - 440020188852978151704071316068 X[2] X[3] X[5] 2 2 + 643931507842423879410406154 X[2] X[3] X[4] 2 - 50521942671324283973813793340 X[2] X[3] X[4] X[5] 2 2 + 617828405654370162075615216122 X[2] X[3] X[5] 3 - 118158641404351483856718996 X[2] X[3] X[4] 2 - 1777376661618017930637914156 X[2] X[3] X[4] X[5] 2 + 43465093209667226161641372900 X[2] X[3] X[4] X[5] 3 - 389811813022876103066510606852 X[2] X[3] X[5] 4 + 661208859994823641414487 X[2] X[4] 3 + 116779417968982321213243564 X[2] X[4] X[5] 2 2 + 1115645584211636053276113722 X[2] X[4] X[5] 3 - 11978327447377906032208842932 X[2] X[4] X[5] 4 + 93420022920703271318131519847 X[2] X[5] 5 - 40631177254845672230995291931 X[3] 4 - 8361841328800225753767201239 X[3] X[4] 4 + 186659909503931963597445749505 X[3] X[5] 3 2 - 53132234392632344591501246 X[3] X[4] 3 + 31043738303811094663009089860 X[3] X[4] X[5] 3 2 - 344513945078470072457164535038 X[3] X[5] 2 3 + 106731726892228699077927234 X[3] X[4] 2 2 + 646487228767090193023715766 X[3] X[4] X[5] 2 2 - 42227138544652118298278943450 X[3] X[4] X[5] 2 3 + 320149353726280071653501215730 X[3] X[5] 4 - 669634127647084603504663 X[3] X[4] 3 - 209007176144195590623869532 X[3] X[4] X[5] 2 2 - 1105095370104993997546647178 X[3] X[4] X[5] 3 + 24827755247784783809017017380 X[3] X[4] X[5] 4 - 150197412365657204550485595031 X[3] X[5] 5 4 - 28105351065649128323675 X[4] + 735876633654616767198257 X[4] X[5] 3 2 + 102712004901569337669582034 X[4] X[5] 2 3 + 511810208017663835685123330 X[4] X[5] 4 - 5284066444066333551668443111 X[4] X[5] 5 + 28531940779179195631486729565 X[5] + 122046302542773526181266432 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j -3 t - 3 t - t - 7 t - 4 ) a[1, j] t = ------------------------------------ / 5 4 3 2 ----- -t + 18 t + 7 t - 9 t + 19 t + 1 j = 0 infinity ----- 4 3 2 \ j -8 t + 15 t - 11 t - 7 t + 13 ) a[2, j] t = ------------------------------------ / 5 4 3 2 ----- -t + 18 t + 7 t - 9 t + 19 t + 1 j = 0 infinity ----- 4 3 2 \ j 12 t + 7 t + 16 t - 8 t + 2 ) a[3, j] t = ------------------------------------ / 5 4 3 2 ----- -t + 18 t + 7 t - 9 t + 19 t + 1 j = 0 infinity ----- 4 3 2 \ j -18 t - 5 t - 7 t + 17 t - 16 ) a[4, j] t = ------------------------------------ / 5 4 3 2 ----- -t + 18 t + 7 t - 9 t + 19 t + 1 j = 0 infinity ----- 4 3 2 \ j 14 t - 7 t + 20 t - 12 t - 13 ) a[5, j] t = ------------------------------------ / 5 4 3 2 ----- -t + 18 t + 7 t - 9 t + 19 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 9 Let P(X[1], X[2], X[3], X[4], X[5]) = -138743 X[1] + 209314 X[2] + 153353 X[3] + 170551 X[4] - 105422 X[5] + 218896 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j -4 t + 15 t + 10 t - 13 t + 14 ) a[1, j] t = ---------------------------------- / 5 4 3 2 ----- -t - 2 t - 9 t + 6 t + 5 t + 1 j = 0 infinity ----- 4 3 2 \ j -4 t - 5 t - t + 3 t + 17 ) a[2, j] t = ---------------------------------- / 5 4 3 2 ----- -t - 2 t - 9 t + 6 t + 5 t + 1 j = 0 infinity ----- 4 3 2 \ j t + 13 t - 18 t - 6 t - 17 ) a[3, j] t = ---------------------------------- / 5 4 3 2 ----- -t - 2 t - 9 t + 6 t + 5 t + 1 j = 0 infinity ----- 4 3 2 \ j -3 t - 4 t + 12 t - 11 t + 7 ) a[4, j] t = ---------------------------------- / 5 4 3 2 ----- -t - 2 t - 9 t + 6 t + 5 t + 1 j = 0 infinity ----- 4 3 2 \ j -4 t - 11 t + 3 t + 9 t + 4 ) a[5, j] t = ---------------------------------- / 5 4 3 2 ----- -t - 2 t - 9 t + 6 t + 5 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 10 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = -8857203153573875092300800000 X[1] 4 - 28080249278339954406026880000 X[1] X[2] 4 - 30617285069439142106126400000 X[1] X[3] 4 + 152731105857898443106453440000 X[1] X[4] 4 - 164008987786430245636554240000 X[1] X[5] 3 2 + 164699207273994115207495356000 X[1] X[2] 3 + 124368882405365401946042220000 X[1] X[2] X[3] 3 - 310333045226417689316468436000 X[1] X[2] X[4] 3 + 199800010651759462299771936000 X[1] X[2] X[5] 3 2 + 11020746385029063201095475000 X[1] X[3] 3 + 55832896385730564200527590000 X[1] X[3] X[4] 3 - 131426389138394548452489840000 X[1] X[3] X[5] 3 2 - 448650143235475603107099021000 X[1] X[4] 3 + 1203785200365930089657250192000 X[1] X[4] X[5] 3 2 - 750938389603962595983948096000 X[1] X[5] 2 3 - 352618373579736002849470807200 X[1] X[2] 2 2 - 408824244285187514341663854000 X[1] X[2] X[3] 2 2 + 482154005458134271169644906800 X[1] X[2] X[4] 2 2 - 47910457752065625233292868800 X[1] X[2] X[5] 2 2 - 152938019008976667450667335000 X[1] X[2] X[3] 2 + 277419984351181214191519374000 X[1] X[2] X[3] X[4] 2 + 43713012187196164644308136000 X[1] X[2] X[3] X[5] 2 2 + 522951237446406114378493038600 X[1] X[2] X[4] 2 - 1660250611718969067609872875200 X[1] X[2] X[4] X[5] 2 2 + 987251688497010874497178521600 X[1] X[2] X[5] 2 3 - 19825494662031324212787412500 X[1] X[3] 2 2 + 88742538314584305330875017500 X[1] X[3] X[4] 2 2 - 50942030151219564958722630000 X[1] X[3] X[5] 2 2 - 191063974249022919697676623500 X[1] X[3] X[4] 2 + 419580186849776364249740292000 X[1] X[3] X[4] X[5] 2 2 - 300765849006562987773027456000 X[1] X[3] X[5] 2 3 + 625896218078486832546403704900 X[1] X[4] 2 2 - 2542589561513100983536069225200 X[1] X[4] X[5] 2 2 + 3432150141502169037432342643200 X[1] X[4] X[5] 2 3 - 1503567847991234890794689510400 X[1] X[5] 4 + 269428841828011125436018347840 X[1] X[2] 3 + 466029158565972151358399127600 X[1] X[2] X[3] 3 - 195301172837912157949170373680 X[1] X[2] X[4] 3 - 389592859717151318228693008320 X[1] X[2] X[5] 2 2 + 314958887619181897692649068000 X[1] X[2] X[3] 2 - 327656948486863932959243936400 X[1] X[2] X[3] X[4] 2 - 385886896762165262304782115600 X[1] X[2] X[3] X[5] 2 2 - 530701641355517510819401128240 X[1] X[2] X[4] 2 + 1744947324102150645992765358480 X[1] X[2] X[4] X[5] 2 2 - 839570886274599662947877478240 X[1] X[2] X[5] 3 + 105386207653962723448049827500 X[1] X[2] X[3] 2 - 240924073376640199483388470500 X[1] X[2] X[3] X[4] 2 - 64255560216736282465526352000 X[1] X[2] X[3] X[5] 2 - 71794474477183368903611004300 X[1] X[2] X[3] X[4] + 680833925221834323325382925600 X[1] X[2] X[3] X[4] X[5] 2 - 312843410889824244713617288800 X[1] X[2] X[3] X[5] 3 - 424421704932635472075834855420 X[1] X[2] X[4] 2 + 1846098935113518632116656185760 X[1] X[2] X[4] X[5] 2 - 2856874066363059924412692689760 X[1] X[2] X[4] X[5] 3 + 1344050347711991458089384241920 X[1] X[2] X[5] 4 + 15628280386718408773204125000 X[1] X[3] 3 - 68295223527135044686468807500 X[1] X[3] X[4] 3 + 17160579020057392444047457500 X[1] X[3] X[5] 2 2 + 40910208439191099356229349500 X[1] X[3] X[4] 2 + 129602305532043341452823593500 X[1] X[3] X[4] X[5] 2 2 - 117181246897491442858796643000 X[1] X[3] X[5] 3 + 91757112885429283215439170300 X[1] X[3] X[4] 2 - 589251937326621510022647543900 X[1] X[3] X[4] X[5] 2 + 823604450925159178804801904400 X[1] X[3] X[4] X[5] 3 - 365143383987411300873185980800 X[1] X[3] X[5] 4 - 386074506411687304107170890260 X[1] X[4] 3 + 2233879434670729268411011773540 X[1] X[4] X[5] 2 2 - 4649319426888213714542457493560 X[1] X[4] X[5] 3 + 4227812879847670119478313255040 X[1] X[4] X[5] 4 - 1419362685062619348976953269760 X[1] X[5] 5 - 63217414370705584608241588832 X[2] 4 - 144233663595496270965245227440 X[2] X[3] 4 - 21830570205299134376224524080 X[2] X[4] 4 + 223302234990345912951335416800 X[2] X[5] 3 2 - 136572711450192656510906053800 X[2] X[3] 3 - 13952189742326738392214224320 X[2] X[3] X[4] 3 + 367161004805786302043782116720 X[2] X[3] X[5] 3 2 + 204503491175848392699112720840 X[2] X[4] 3 - 473570653688599818395931790320 X[2] X[4] X[5] 3 2 + 50642955469955883811182598080 X[2] X[5] 2 3 - 69428782380291640621916193000 X[2] X[3] 2 2 + 18449239307719153223074939200 X[2] X[3] X[4] 2 2 + 228291549553136760079099069800 X[2] X[3] X[5] 2 2 + 242165667328377414609414635640 X[2] X[3] X[4] 2 - 624176995965205839633159788880 X[2] X[3] X[4] X[5] 2 2 + 103448748659646934790059368240 X[2] X[3] X[5] 2 3 + 116196634772291011445531308400 X[2] X[4] 2 2 - 803544782013475809159088111320 X[2] X[4] X[5] 2 2 + 1442948329501609116686598941040 X[2] X[4] X[5] 2 3 - 656491456934191553611824321120 X[2] X[5] 4 - 19376410120919545896091147500 X[2] X[3] 3 + 16454034177670043641417315500 X[2] X[3] X[4] 3 + 68894240102260134442720204500 X[2] X[3] X[5] 2 2 + 127009420440016904204327784900 X[2] X[3] X[4] 2 - 374031170839549889005522506300 X[2] X[3] X[4] X[5] 2 2 + 118857780409667455289411906400 X[2] X[3] X[5] 3 - 82271335147947266666098551180 X[2] X[3] X[4] 2 + 31628011176795783881168399340 X[2] X[3] X[4] X[5] 2 + 352583322280230757558387167360 X[2] X[3] X[4] X[5] 3 - 218459855653094049695307335520 X[2] X[3] X[5] 4 + 156547950369862542552776104520 X[2] X[4] 3 - 780769220328900806159224334340 X[2] X[4] X[5] 2 2 + 1618537086625459055807348589600 X[2] X[4] X[5] 3 - 1613661112303697188618843793760 X[2] X[4] X[5] 4 + 599796274991167852806330996480 X[2] X[5] 5 - 2356967353971105206926903125 X[3] 4 + 2824024522787307039948620625 X[3] X[4] 4 + 9774219621481712762164470000 X[3] X[5] 3 2 + 34771026592441236058306296750 X[3] X[4] 3 - 101616776766634376495657466000 X[3] X[4] X[5] 3 2 + 43893975621451069281833275500 X[3] X[5] 2 3 - 67353531458926870391243286150 X[3] X[4] 2 2 + 139393486933016201207545768200 X[3] X[4] X[5] 2 2 - 8308965439296950980698326700 X[3] X[4] X[5] 2 3 - 45529905379069069979455236600 X[3] X[5] 4 + 25480767386921378993760958935 X[3] X[4] 3 + 74675111452060151669364552960 X[3] X[4] X[5] 2 2 - 415203273413484140384957203740 X[3] X[4] X[5] 3 + 473306010404079756377840985360 X[3] X[4] X[5] 4 - 166731082847033028297147296640 X[3] X[5] 5 + 77495626905699398006630787941 X[4] 4 - 647737892982395603064583034760 X[4] X[5] 3 2 + 1943828261344102614880267485180 X[4] X[5] 2 3 - 2760158683500092623618544699640 X[4] X[5] 4 + 1905944895544714744096375171200 X[4] X[5] 5 - 518016776166834071703113719296 X[5] + 5906268855853531515669800700000 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j -18 t - 2 t + 5 t - 16 t - 14 ) a[1, j] t = ---------------------------------- / 5 4 3 2 ----- -t - 6 t + 2 t - 9 t - 7 t + 1 j = 0 infinity ----- 3 2 \ j 9 t + 20 t - t - 5 ) a[2, j] t = ---------------------------------- / 5 4 3 2 ----- -t - 6 t + 2 t - 9 t - 7 t + 1 j = 0 infinity ----- 4 3 2 \ j -18 t - 20 t + 4 t + 17 t + 12 ) a[3, j] t = ---------------------------------- / 5 4 3 2 ----- -t - 6 t + 2 t - 9 t - 7 t + 1 j = 0 infinity ----- 4 2 \ j -12 t + 14 t + 5 t + 16 ) a[4, j] t = ---------------------------------- / 5 4 3 2 ----- -t - 6 t + 2 t - 9 t - 7 t + 1 j = 0 infinity ----- 4 3 2 \ j 3 t - 3 t + 14 t + 12 t + 16 ) a[5, j] t = ---------------------------------- / 5 4 3 2 ----- -t - 6 t + 2 t - 9 t - 7 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 11 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 3111432272759771178584825000 X[1] 4 + 6906875953831756078069905000 X[1] X[2] 4 - 17430288299881163145647730000 X[1] X[3] 4 + 60078593018163990246460240000 X[1] X[4] 4 - 40247503949811957637869817500 X[1] X[5] 3 2 + 2152608468468361458679001250 X[1] X[2] 3 - 37701849739549238393536372500 X[1] X[2] X[3] 3 + 83379388482235038474709878750 X[1] X[2] X[4] 3 - 32183182775726437019230957500 X[1] X[2] X[5] 3 2 + 37599005185812501620750021250 X[1] X[3] 3 - 311161440353413816334947601250 X[1] X[3] X[4] 3 + 243901772810214110420719590000 X[1] X[3] X[5] 3 2 + 437763614071039632471656918750 X[1] X[4] 3 - 488517850834493629255128243750 X[1] X[4] X[5] 3 2 + 66890542341574408890100162500 X[1] X[5] 2 3 - 1395940658509197707853608775 X[1] X[2] 2 2 - 9122075810118265763888798175 X[1] X[2] X[3] 2 2 + 5068197598261818617532051525 X[1] X[2] X[4] 2 2 + 8820545653568006499111694200 X[1] X[2] X[5] 2 2 + 97679748327041409732517190175 X[1] X[2] X[3] 2 - 381861857435748982386905233800 X[1] X[2] X[3] X[4] 2 + 115358493399963877821751928100 X[1] X[2] X[3] X[5] 2 2 + 361077530631075669014055799200 X[1] X[2] X[4] 2 - 200757779430885122341459010175 X[1] X[2] X[4] X[5] 2 2 + 23041313271863025317242889175 X[1] X[2] X[5] 2 3 - 29706142637292823082168433225 X[1] X[3] 2 2 + 649447867270750427158318672275 X[1] X[3] X[4] 2 2 - 642308058611827244257307914800 X[1] X[3] X[5] 2 2 - 1923166319436359552046092733450 X[1] X[3] X[4] 2 + 2357071172225168420401966674675 X[1] X[3] X[4] X[5] 2 2 - 263225223432732294250493699925 X[1] X[3] X[5] 2 3 + 1562324604990368857909145000075 X[1] X[4] 2 2 - 2213593877456928932420015265825 X[1] X[4] X[5] 2 2 + 525356229614652577491677312400 X[1] X[4] X[5] 2 3 - 45604904229670086986124858975 X[1] X[5] 4 + 202958875009976016246446160 X[1] X[2] 3 + 4966020987328475857659141585 X[1] X[2] X[3] 3 - 6168663290165399295660629505 X[1] X[2] X[4] 3 - 791709263885762434756076265 X[1] X[2] X[5] 2 2 + 22567485422318332660691808285 X[1] X[2] X[3] 2 - 28625558163674811268820088960 X[1] X[2] X[3] X[4] 2 - 35653353055108196216151550005 X[1] X[2] X[3] X[5] 2 2 - 5011150593501982531540241685 X[1] X[2] X[4] 2 + 54291448104908580341330263515 X[1] X[2] X[4] X[5] 2 2 - 3710262573193540568578502040 X[1] X[2] X[5] 3 - 132047853003930884568573783465 X[1] X[2] X[3] 2 + 750401467015625395581593942685 X[1] X[2] X[3] X[4] 2 - 214464686323539444072218976195 X[1] X[2] X[3] X[5] 2 - 1309104004480928824517272890180 X[1] X[2] X[3] X[4] + 589347473707072880207489146545 X[1] X[2] X[3] X[4] X[5] 2 - 12265322253172756759813723245 X[1] X[2] X[3] X[5] 3 + 738188470091353897177300828680 X[1] X[2] X[4] 2 - 442243082509753802998468031505 X[1] X[2] X[4] X[5] 2 + 51469810614182324735519931360 X[1] X[2] X[4] X[5] 3 - 10946238981995760952857483015 X[1] X[2] X[5] 4 - 25375663445942888156161262565 X[1] X[3] 3 - 584149201266025453938434834220 X[1] X[3] X[4] 3 + 904717119673097811236145747465 X[1] X[3] X[5] 2 2 + 3148982463247637043517759433115 X[1] X[3] X[4] 2 - 4612832014307179016394020235060 X[1] X[3] X[4] X[5] 2 2 + 394060332259360556816703251535 X[1] X[3] X[5] 3 - 5211493737338678522207425290855 X[1] X[3] X[4] 2 + 7918435812934517216197626483930 X[1] X[3] X[4] X[5] 2 - 1415600519610840159114299121210 X[1] X[3] X[4] X[5] 3 + 96173372048155497079133047290 X[1] X[3] X[5] 4 + 2853281327960768153712616022485 X[1] X[4] 3 - 4630111683216503357772361328430 X[1] X[4] X[5] 2 2 + 1344369419821703304479136989565 X[1] X[4] X[5] 3 - 211615889246612996994111562245 X[1] X[4] X[5] 4 + 16097581544115710672522812035 X[1] X[5] 5 4 - 8941227753105393128780688 X[2] - 390436112572046826531276780 X[2] X[3] 4 + 518179656086745144903442620 X[2] X[4] 4 + 17765858994095152957607820 X[2] X[5] 3 2 - 5632136021553073795137185325 X[2] X[3] 3 + 13351983663483660444413197680 X[2] X[3] X[4] 3 + 2080988882904110182964731800 X[2] X[3] X[5] 3 2 - 7725493964505967839639694215 X[2] X[4] 3 - 3017312025907784429870670870 X[2] X[4] X[5] 3 2 + 172031237883237164360947425 X[2] X[5] 2 3 - 26380170303449590136867819715 X[2] X[3] 2 2 + 64535758829518449772892774565 X[2] X[3] X[4] 2 2 + 42590272602205950342387846030 X[2] X[3] X[5] 2 2 - 29993393852688391650216282285 X[2] X[3] X[4] 2 - 118329674091371349670902260070 X[2] X[3] X[4] X[5] 2 2 + 3432871010556297024684051555 X[2] X[3] X[5] 2 3 - 13262455174720374408231686925 X[2] X[4] 2 2 + 84913137656065069080488076240 X[2] X[4] X[5] 2 2 - 7165160288530994914883423685 X[2] X[4] X[5] 2 3 + 775743591930652204306349310 X[2] X[5] 4 + 67797540054974068129486308465 X[2] X[3] 3 - 566485484250546217684781697150 X[2] X[3] X[4] 3 + 202767090816796010587492165260 X[2] X[3] X[5] 2 2 + 1493781623902492842284508044565 X[2] X[3] X[4] 2 - 723473148196953867436429016025 X[2] X[3] X[4] X[5] 2 2 - 30042083638565634123923017860 X[2] X[3] X[5] 3 - 1619011900173785518182285212310 X[2] X[3] X[4] 2 + 880083301548522733639138871280 X[2] X[3] X[4] X[5] 2 + 55514604436439802085774631250 X[2] X[3] X[4] X[5] 3 + 6238518963897227646753900360 X[2] X[3] X[5] 4 + 633088315953675275025567515730 X[2] X[4] 3 - 374356259144003740258943592885 X[2] X[4] X[5] 2 2 - 10421444489093186883553070685 X[2] X[4] X[5] 3 - 15288515332438893175705071375 X[2] X[4] X[5] 4 + 1737415907785504939498815015 X[2] X[5] 5 + 47563440934995318610266779043 X[3] 4 + 99285810940022244892588114785 X[3] X[4] 4 - 546510179908402896507504065910 X[3] X[5] 3 2 - 1750505472994983928745354588565 X[3] X[4] 3 + 3591456388259241012207902492715 X[3] X[4] X[5] 3 2 - 258552008206849390110611534370 X[3] X[5] 2 3 + 4883610183269267801256586440210 X[3] X[4] 2 2 - 8722148397752041783189604923995 X[3] X[4] X[5] 2 2 + 1193648669923779553961712305385 X[3] X[4] X[5] 2 3 - 42777033115472470587893863095 X[3] X[5] 4 - 5424248262081186506873412045450 X[3] X[4] 3 + 9431122342307128160861104770195 X[3] X[4] X[5] 2 2 - 1975586629182734769573926906145 X[3] X[4] X[5] 3 + 202093247987563697202742538265 X[3] X[4] X[5] 4 - 16669991060246990437846190685 X[3] X[5] 5 + 2191263839420793867250990123181 X[4] 4 - 3879627359002821758978251632975 X[4] X[5] 3 2 + 1167138325963595958175619163585 X[4] X[5] 2 3 - 228167116775627377891459772340 X[4] X[5] 4 + 36189982887807574332295212060 X[4] X[5] 5 - 2187020095086078283022529507 X[5] + 276734276011811424969917634375 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j 15 t - 12 t - 11 t + 11 t - 6 ) a[1, j] t = ------------------------------------- / 5 4 3 2 ----- -t - 19 t + 12 t + 10 t + 4 t + 1 j = 0 infinity ----- 4 3 2 \ j 17 t + 7 t + 18 t + 19 t - 12 ) a[2, j] t = ------------------------------------- / 5 4 3 2 ----- -t - 19 t + 12 t + 10 t + 4 t + 1 j = 0 infinity ----- 4 3 2 \ j -4 t + 9 t + 19 t - 5 t + 19 ) a[3, j] t = ------------------------------------- / 5 4 3 2 ----- -t - 19 t + 12 t + 10 t + 4 t + 1 j = 0 infinity ----- 4 3 2 \ j -6 t + 15 t + 14 t - 8 t + 15 ) a[4, j] t = ------------------------------------- / 5 4 3 2 ----- -t - 19 t + 12 t + 10 t + 4 t + 1 j = 0 infinity ----- 4 3 2 \ j 13 t + 3 t + 6 t - 5 t + 9 ) a[5, j] t = ------------------------------------- / 5 4 3 2 ----- -t - 19 t + 12 t + 10 t + 4 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 12 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 994320269996340080507445533088 X[1] 4 - 10484030671562406970156905045632 X[1] X[2] 4 - 1066169379695554293218454049888 X[1] X[3] 4 + 16669080270014163530309131710592 X[1] X[4] 4 + 11076244994951041700199232558464 X[1] X[5] 3 2 + 46920145582370106505211947620576 X[1] X[2] 3 + 8392344729466071561917226056352 X[1] X[2] X[3] 3 - 152574460183494524562534125740016 X[1] X[2] X[4] 3 - 94365643030090677100033169583952 X[1] X[2] X[5] 3 2 + 174940608274262920355460924576 X[1] X[3] 3 - 12582623340815636542853271760384 X[1] X[3] X[4] 3 - 7990526693454216890892240322784 X[1] X[3] X[5] 3 2 + 124633354638837219912085925809960 X[1] X[4] 3 + 149283128025681669796852611541136 X[1] X[4] X[5] 3 2 + 47433031398209438662380554898408 X[1] X[5] 2 3 - 107334062617827947141337586461280 X[1] X[2] 2 2 - 24450390344377222923371142352160 X[1] X[2] X[3] 2 2 + 526424203049546255550300661149536 X[1] X[2] X[4] 2 2 + 310258423377694808009467677056960 X[1] X[2] X[5] 2 2 - 653484574354896233984459829920 X[1] X[2] X[3] 2 + 72366293347438147894084903881488 X[1] X[2] X[3] X[4] 2 + 45297176123560229672900723179504 X[1] X[2] X[3] X[5] 2 2 - 858760279950905481497299395709744 X[1] X[2] X[4] 2 - 990570843569505598005937160818336 X[1] X[2] X[4] X[5] 2 2 - 299631239106586377360942576766576 X[1] X[2] X[5] 2 3 + 120401733411238202207562638528 X[1] X[3] 2 2 + 478099878862599106593704646368 X[1] X[3] X[4] 2 2 + 59047105737380504318855502336 X[1] X[3] X[5] 2 2 - 52647579897504875008839067917264 X[1] X[3] X[4] 2 - 64859827585012865130389489701824 X[1] X[3] X[4] X[5] 2 2 - 19794631277991045406575997542416 X[1] X[3] X[5] 2 3 + 464549056122320809459278551405996 X[1] X[4] 2 2 + 789383290681976456293853433021420 X[1] X[4] X[5] 2 2 + 465502018984684834257421895628116 X[1] X[4] X[5] 2 3 + 95318511342145304663805481762932 X[1] X[5] 4 + 122641532636521204147863618312864 X[1] X[2] 3 + 30919392485135585809631875060960 X[1] X[2] X[3] 3 - 798716667266030391809199397732704 X[1] X[2] X[4] 3 - 456602111773290779853715292773888 X[1] X[2] X[5] 2 2 + 116050707438694784447003109184 X[1] X[2] X[3] 2 - 133873375282334625348294914176144 X[1] X[2] X[3] X[4] 2 - 82787976369710048807127884539792 X[1] X[2] X[3] X[5] 2 2 + 1937655846212859502585810427837928 X[1] X[2] X[4] 2 + 2182686554609837990680196665850960 X[1] X[2] X[4] X[5] 2 2 + 637400676902080475195262032984552 X[1] X[2] X[5] 3 - 553959115200330188693736905952 X[1] X[2] X[3] 2 + 2300664379126598491769705253280 X[1] X[2] X[3] X[4] 2 + 1693383761040181038991730712832 X[1] X[2] X[3] X[5] 2 + 187717929544710909048275163861560 X[1] X[2] X[3] X[4] + 229719352104621576986632575622496 X[1] X[2] X[3] X[4] X[5] 2 + 70003120000769390679636911901320 X[1] X[2] X[3] X[5] 3 - 2069841960524665791571729457418892 X[1] X[2] X[4] 2 - 3452950391117185470430253455426276 X[1] X[2] X[4] X[5] 2 - 1979510394942361300061751850531348 X[1] X[2] X[4] X[5] 3 - 391541407933041488794357185224252 X[1] X[2] X[5] 4 - 31698986256423452502139598048 X[1] X[3] 3 + 902488566784361464002371974016 X[1] X[3] X[4] 3 + 604246526668207656472270451968 X[1] X[3] X[5] 2 2 - 3975230433697826859541146265936 X[1] X[3] X[4] 2 - 4845101808550165751921551247696 X[1] X[3] X[4] X[5] 2 2 - 1848497464031699859147460344160 X[1] X[3] X[5] 3 - 84092217365944696346272256040052 X[1] X[3] X[4] 2 - 153421649235505418927996780740564 X[1] X[3] X[4] X[5] 2 - 92455796523047236138125057482396 X[1] X[3] X[4] X[5] 3 - 17997482101593289033784641762972 X[1] X[3] X[5] 4 + 819027561207209807662297740571550 X[1] X[4] 3 + 1801534197836409832817070865358264 X[1] X[4] X[5] 2 2 + 1524092559789214561371894706392252 X[1] X[4] X[5] 3 + 589776871249807877508923093098936 X[1] X[4] X[5] 4 + 87876732791157789809530093219350 X[1] X[5] 5 - 55365718261458247574750223399328 X[2] 4 - 14207995714015769962181165016160 X[2] X[3] 4 + 446809745669901766140833050052240 X[2] X[4] 4 + 250686368276341762799105796501520 X[2] X[5] 3 2 + 855745287571804502738228179744 X[2] X[3] 3 + 79330975454191226652564739932288 X[2] X[3] X[4] 3 + 48653246858605467199471892678944 X[2] X[3] X[5] 3 2 - 1428897346702407581888610301461976 X[2] X[4] 3 - 1586051066342914727508253609875872 X[2] X[4] X[5] 3 2 - 452006315856091330029037752240232 X[2] X[5] 2 3 + 592307444210753853568450723872 X[2] X[3] 2 2 - 6495108904912657412529394593232 X[2] X[3] X[4] 2 2 - 3566557657306059376156449550736 X[2] X[3] X[5] 2 2 - 160201597979976494609912190400544 X[2] X[3] X[4] 2 - 195928624601286717263598109353312 X[2] X[3] X[4] X[5] 2 2 - 59971574710425951780640513187712 X[2] X[3] X[5] 2 3 + 2258857549671187192235306206727948 X[2] X[4] 2 2 + 3724357765581540758851228442500676 X[2] X[4] X[5] 2 2 + 2093231996461787303919891932001668 X[2] X[4] X[5] 2 3 + 403176322095484215879016978726188 X[2] X[5] 4 + 53444134385132376697955735936 X[2] X[3] 3 - 1782196046435815504471383796336 X[2] X[3] X[4] 3 - 1142274036423015338572010200240 X[2] X[3] X[5] 2 2 + 13313582029206326256398834501200 X[2] X[3] X[4] 2 + 14267522314658600270203616899920 X[2] X[3] X[4] X[5] 2 2 + 4552409190616465986544558230336 X[2] X[3] X[5] 3 + 137492900504345213275224508111940 X[2] X[3] X[4] 2 + 253073667478505403357307418940204 X[2] X[3] X[4] X[5] 2 + 154195707127992626098835447674924 X[2] X[3] X[4] X[5] 3 + 30313968435293630113129889792260 X[2] X[3] X[5] 4 - 1761441439865512116775682607694928 X[2] X[4] 3 - 3837411935291998202243886127277232 X[2] X[4] X[5] 2 2 - 3193888945704652623527003870850200 X[2] X[4] X[5] 3 - 1208935000738164373230575037892912 X[2] X[4] X[5] 4 - 175651771258147424546469254356120 X[2] X[5] 5 - 519395188044699692863032928 X[3] 4 - 52097824746218807358114170896 X[3] X[4] 4 - 19001517001162297158924281648 X[3] X[5] 3 2 + 1206036040776736928741770941368 X[3] X[4] 3 + 1428342322647432096019275759904 X[3] X[4] X[5] 3 2 + 383700980103121132103483310472 X[3] X[5] 2 3 - 7831467144022946838439672845652 X[3] X[4] 2 2 - 12242996585400317522433142824924 X[3] X[4] X[5] 2 2 - 7136809390145240028880389779340 X[3] X[4] X[5] 2 3 - 1427247351672189836329198788100 X[3] X[5] 4 - 42190734012214071937510970737824 X[3] X[4] 3 - 104679622714852601285607713743548 X[3] X[4] X[5] 2 2 - 95743906486372664023054829863820 X[3] X[4] X[5] 3 - 37511851989863529363948365915764 X[3] X[4] X[5] 4 - 5294280458800151195525767735076 X[3] X[5] 5 + 540977766399908742296398006585381 X[4] 4 + 1460281426415872792263443995752625 X[4] X[5] 3 2 + 1600862620956167783646536145342586 X[4] X[5] 2 3 + 894045143462402654765696340279026 X[4] X[5] 4 + 254390419571549856593415278162945 X[4] X[5] 5 + 29386822669049004424197063546909 X[5] + 138762723978624488782333796278624 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j -4 t - 18 t + 19 t + 18 t - 19 ) a[1, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 19 t + 2 t + 17 t + 9 t + 1 j = 0 infinity ----- 4 3 2 \ j 17 t + 4 t - 7 t - t - 20 ) a[2, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 19 t + 2 t + 17 t + 9 t + 1 j = 0 infinity ----- 4 3 2 \ j 19 t + 14 t + 20 t + 20 t + 11 ) a[3, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 19 t + 2 t + 17 t + 9 t + 1 j = 0 infinity ----- 4 3 2 \ j 6 t + 9 t - 12 t - 2 t - 10 ) a[4, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 19 t + 2 t + 17 t + 9 t + 1 j = 0 infinity ----- 4 3 2 \ j 10 t + 11 t + 4 t - 12 t + 4 ) a[5, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 19 t + 2 t + 17 t + 9 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 13 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 84075198855675773512723399 X[1] 4 + 382272189312967409102203701 X[1] X[2] 4 + 140181292931961891936623748 X[1] X[3] 4 + 18536496694272460924845573 X[1] X[4] 4 + 32754729677104978595276225 X[1] X[5] 3 2 - 161869413261438768882163310 X[1] X[2] 3 - 70314084512129287647036272 X[1] X[2] X[3] 3 - 3147012182602966861085493228 X[1] X[2] X[4] 3 + 3320220105922233059028487492 X[1] X[2] X[5] 3 2 + 373619339879914058906607568 X[1] X[3] 3 + 1811556680725672717724437536 X[1] X[3] X[4] 3 - 2053519085214885979366024224 X[1] X[3] X[5] 3 2 + 2474532085493278741647661846 X[1] X[4] 3 - 5840254008222318457328105292 X[1] X[4] X[5] 3 2 + 3439631678557193109968693602 X[1] X[5] 2 3 + 41100742962035329674584263622 X[1] X[2] 2 2 - 39554645532766573590604872568 X[1] X[2] X[3] 2 2 - 75955013350700469532777573966 X[1] X[2] X[4] 2 2 + 108159806275306681602938082634 X[1] X[2] X[5] 2 2 + 11161590924774598061530150576 X[1] X[2] X[3] 2 + 40368440733458382654233634528 X[1] X[2] X[3] X[4] 2 - 60714701237693797613418708064 X[1] X[2] X[3] X[5] 2 2 + 46763193797256817822798603542 X[1] X[2] X[4] 2 - 131386990075259893176204632716 X[1] X[2] X[4] X[5] 2 2 + 93014003265166947491715676466 X[1] X[2] X[5] 2 3 - 631578718677462196465513536 X[1] X[3] 2 2 - 3248114933323891203731367472 X[1] X[3] X[4] 2 2 + 5914917946722407645377279984 X[1] X[3] X[5] 2 2 - 8492292240591418122521560568 X[1] X[3] X[4] 2 + 26770099372844123086050875376 X[1] X[3] X[4] X[5] 2 2 - 20839956544298677453784259560 X[1] X[3] X[5] 2 3 - 8326354236039679148279591270 X[1] X[4] 2 2 + 36351706099450862683488295790 X[1] X[4] X[5] 2 2 - 52464153047042724315136798542 X[1] X[4] X[5] 2 3 + 25154872769872350227015881582 X[1] X[5] 4 - 146507075641534366019333891353 X[1] X[2] 3 + 262216847310987524675185241424 X[1] X[2] X[3] 3 + 369970891077394059749798391148 X[1] X[2] X[4] 3 - 507351138071855532463431776004 X[1] X[2] X[5] 2 2 - 161738745835605660770639323920 X[1] X[2] X[3] 2 - 498678886278168987804402237696 X[1] X[2] X[3] X[4] 2 + 684805014966634468139430683744 X[1] X[2] X[3] X[5] 2 2 - 334232556314534642024971270154 X[1] X[2] X[4] 2 + 922056797874589382772956857156 X[1] X[2] X[4] X[5] 2 2 - 635689004818195462278799343654 X[1] X[2] X[5] 3 + 40678615966831117862284929280 X[1] X[2] X[3] 2 + 198897399020381564055755692320 X[1] X[2] X[3] X[4] 2 - 275283440238306027077873447712 X[1] X[2] X[3] X[5] 2 + 300580382026089097378985298448 X[1] X[2] X[3] X[4] - 831589436478963986261918793888 X[1] X[2] X[3] X[4] X[5] 2 + 574552149185941060609538426320 X[1] X[2] X[3] X[5] 3 + 125180160779748208276138891460 X[1] X[2] X[4] 2 - 527471594371620154065911464340 X[1] X[2] X[4] X[5] 2 + 735829688737337389308081088996 X[1] X[2] X[4] X[5] 3 - 340885300411801865262628713636 X[1] X[2] X[5] 4 - 3406163983074763644026141696 X[1] X[3] 3 - 23118166026921078329099312256 X[1] X[3] X[4] 3 + 32550416077164787424951692800 X[1] X[3] X[5] 2 2 - 56521382410740678843122089232 X[1] X[3] X[4] 2 + 158685877293089172599832772000 X[1] X[3] X[4] X[5] 2 2 - 111024606102990060133460283664 X[1] X[3] X[5] 3 - 55482753569580813338653363840 X[1] X[3] X[4] 2 + 235619021236682026172798037248 X[1] X[3] X[4] X[5] 2 - 330779530224407473433541371040 X[1] X[3] X[4] X[5] 3 + 153985040725861434529325534656 X[1] X[3] X[5] 4 - 15210805432707195538332366957 X[1] X[4] 3 + 90353170031819715135851876356 X[1] X[4] X[5] 2 2 - 196463090727073673738155739258 X[1] X[4] X[5] 3 + 186748091948375488462521673356 X[1] X[4] X[5] 4 - 65844132575456833936493739113 X[1] X[5] 5 + 258451639025756456482024651237 X[2] 4 - 602114981625711021708181912172 X[2] X[3] 4 - 600620809107825855857524330839 X[2] X[4] 4 + 972359251451126645410639454069 X[2] X[5] 3 2 + 550103026481845902890734539664 X[2] X[3] 3 + 1186171613519656902602162715488 X[2] X[3] X[4] 3 - 1856185134882161929254685967360 X[2] X[3] X[5] 3 2 + 500904839339828280894784263998 X[2] X[4] 3 - 1679396575830235004541778119404 X[2] X[4] X[5] 3 2 + 1397689574277054414747866044530 X[2] X[5] 2 3 - 243619753885699951145065349056 X[2] X[3] 2 2 - 847792838153386006946282811056 X[2] X[3] X[4] 2 2 + 1294509169416175077538049281136 X[2] X[3] X[5] 2 2 - 808405429556038814665645451400 X[2] X[3] X[4] 2 + 2582851327099438696360326427184 X[2] X[3] X[4] X[5] 2 2 - 2060212063569470400751468199800 X[2] X[3] X[5] 2 3 - 205743705456873697395511270394 X[2] X[4] 2 2 + 1008227229500686030616691129154 X[2] X[4] X[5] 2 2 - 1700499691048999385844996596618 X[2] X[4] X[5] 2 3 + 963387985052869885133521822154 X[2] X[5] 4 + 51859475701697869592033769472 X[2] X[3] 3 + 255655629392985422421054836224 X[2] X[3] X[4] 3 - 384527112199267533265972051584 X[2] X[3] X[5] 2 2 + 407636393507703804524347791920 X[2] X[3] X[4] 2 - 1262377493999766827855449701728 X[2] X[3] X[4] X[5] 2 2 + 977834429571708915696218117424 X[2] X[3] X[5] 3 + 237822760009854048691910247552 X[2] X[3] X[4] 2 - 1126348420873135284419600560128 X[2] X[3] X[4] X[5] 2 + 1811592416792897329153361448000 X[2] X[3] X[4] X[5] 3 - 977895576550112997345878232480 X[2] X[3] X[5] 4 + 57131364146945182142383943645 X[2] X[4] 3 - 310874569374313461890993236132 X[2] X[4] X[5] 2 2 + 703802836592965389057953553722 X[2] X[4] X[5] 3 - 764464112468320969404807481004 X[2] X[4] X[5] 4 + 324176278124862329878161059481 X[2] X[5] 5 - 4202809939422917090940603392 X[3] 4 - 27111934522765484289656241152 X[3] X[4] 4 + 40535130649302927098862672896 X[3] X[5] 3 2 - 62636409471941981949256412992 X[3] X[4] 3 + 191065097490818842693882760448 X[3] X[4] X[5] 3 2 - 145695353068751898708441715392 X[3] X[5] 2 3 - 62781025553952353566759469200 X[3] X[4] 2 2 + 290906763716980533520901984368 X[3] X[4] X[5] 2 2 - 454002002497347820077625514416 X[3] X[4] X[5] 2 3 + 237300940943268401677470575696 X[3] X[5] 4 - 29049434246119572767382860844 X[3] X[4] 3 + 169784854009577593622789616944 X[3] X[4] X[5] 2 2 - 391546355605064751406361252728 X[3] X[4] X[5] 3 + 416076422558864195449984288368 X[3] X[4] X[5] 4 - 169227460460253120209248129980 X[3] X[5] 5 - 8459163826170598591327723279 X[4] 4 + 49822028686146144513765401369 X[4] X[5] 3 2 - 125205637890192269737828477018 X[4] X[5] 2 3 + 173036220054829869096393590518 X[4] X[5] 4 - 132095080618587627326101428919 X[4] X[5] 5 + 43483862577621548125381972913 X[5] + 3803074578342113945096979500032 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j -t - 7 t + 18 t - 9 t + 6 ) a[1, j] t = ---------------------------------- / 5 4 3 2 ----- -t + t - 3 t - 17 t - 12 t + 1 j = 0 infinity ----- 4 3 2 \ j 4 t - t + 7 t + 9 t - 17 ) a[2, j] t = ---------------------------------- / 5 4 3 2 ----- -t + t - 3 t - 17 t - 12 t + 1 j = 0 infinity ----- 4 3 2 \ j -3 t + 20 t + 16 t + 20 t - 19 ) a[3, j] t = ---------------------------------- / 5 4 3 2 ----- -t + t - 3 t - 17 t - 12 t + 1 j = 0 infinity ----- 4 3 2 \ j -15 t - t - 6 t + 11 t + 12 ) a[4, j] t = ---------------------------------- / 5 4 3 2 ----- -t + t - 3 t - 17 t - 12 t + 1 j = 0 infinity ----- 4 3 2 \ j -18 t + 11 t - 15 t - 13 t + 19 ) a[5, j] t = ---------------------------------- / 5 4 3 2 ----- -t + t - 3 t - 17 t - 12 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 14 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 5807933223901302853400137435 X[1] 4 + 13682638265297854556725352752 X[1] X[2] 4 - 13715189847288670858511942826 X[1] X[3] 4 + 43812146523967907847979175932 X[1] X[4] 4 - 17383370863305695999250970420 X[1] X[5] 3 2 + 3829929425843330961861762796 X[1] X[2] 3 - 13132798258818313608847480004 X[1] X[2] X[3] 3 + 94099260034563278911880780992 X[1] X[2] X[4] 3 - 40422396019575667622667993448 X[1] X[2] X[5] 3 2 + 8210793126623551358973359828 X[1] X[3] 3 - 90873287842761915337671992614 X[1] X[3] X[4] 3 + 38822171430922516423841642694 X[1] X[3] X[5] 3 2 + 127922715021567478929484336736 X[1] X[4] 3 - 99527777242416260693791006538 X[1] X[4] X[5] 3 2 + 18934715759176929114099738330 X[1] X[5] 2 3 - 1806325083837951399276115112 X[1] X[2] 2 2 + 306927866618963868535822416 X[1] X[2] X[3] 2 2 + 23816672368486039341707164196 X[1] X[2] X[4] 2 2 - 9924535185504607115392496308 X[1] X[2] X[5] 2 2 + 3636156512629476310790759440 X[1] X[2] X[3] 2 - 70713124114505304185975253848 X[1] X[2] X[3] X[4] 2 + 30317502304147017665014894840 X[1] X[2] X[3] X[5] 2 2 + 239314158534267838544002817196 X[1] X[2] X[4] 2 - 206605186908509411274349512200 X[1] X[2] X[4] X[5] 2 2 + 44670436348751765441365588132 X[1] X[2] X[5] 2 3 - 2052000127305786412617158280 X[1] X[3] 2 2 + 41707048158811307862471283692 X[1] X[3] X[4] 2 2 - 18171346072904529543453707548 X[1] X[3] X[5] 2 2 - 223036213949645456161564543608 X[1] X[3] X[4] 2 + 190887733899376960155377645652 X[1] X[3] X[4] X[5] 2 2 - 40723567826910773064299759876 X[1] X[3] X[5] 2 3 + 178657187192741041433267221703 X[1] X[4] 2 2 - 201861908979805435090368930187 X[1] X[4] X[5] 2 2 + 73287540981477576032734566117 X[1] X[4] X[5] 2 3 - 8397263769254716470039394985 X[1] X[5] 4 - 10068120138272896550813360 X[1] X[2] 3 + 498071473706481513622591312 X[1] X[2] X[3] 3 - 5942928044724101144573713248 X[1] X[2] X[4] 3 + 2678920014702919473808317328 X[1] X[2] X[5] 2 2 - 78331279334110101506825328 X[1] X[2] X[3] 2 - 778628841071205900114182736 X[1] X[2] X[3] X[4] 2 + 245322211744771279475875808 X[1] X[2] X[3] X[5] 2 2 + 47696338365667888924104494164 X[1] X[2] X[4] 2 - 40335561354482364495063283152 X[1] X[2] X[4] X[5] 2 2 + 8535941831328205969845001180 X[1] X[2] X[5] 3 - 548557891795952692506690848 X[1] X[2] X[3] 2 + 13267931056775784511370896192 X[1] X[2] X[3] X[4] 2 - 5810285299590259781051671872 X[1] X[2] X[3] X[5] 2 - 124555803468583047284865238836 X[1] X[2] X[3] X[4] + 107396318215223305456092314192 X[1] X[2] X[3] X[4] X[5] 2 - 23165787516014646867075962044 X[1] X[2] X[3] X[5] 3 + 266129934484653062575978650526 X[1] X[2] X[4] 2 - 346155326506568233254222796406 X[1] X[2] X[4] X[5] 2 + 150315434136527436305056508314 X[1] X[2] X[4] X[5] 3 - 21797640545651566520094614706 X[1] X[2] X[5] 4 + 236219970953386434663769472 X[1] X[3] 3 - 6822120561225732893984276048 X[1] X[3] X[4] 3 + 3049286136416449902982023648 X[1] X[3] X[5] 2 2 + 69589446143090788318932412552 X[1] X[3] X[4] 2 - 60864546793857607516189064536 X[1] X[3] X[4] X[5] 2 2 + 13293104369098255406864946272 X[1] X[3] X[5] 3 - 240153302276986376522186048350 X[1] X[3] X[4] 2 + 308750982535259935140897549678 X[1] X[3] X[4] X[5] 2 - 131868872485547455216753282786 X[1] X[3] X[4] X[5] 3 + 18707161921244708136208465938 X[1] X[3] X[5] 4 + 117100654546485341015659392424 X[1] X[4] 3 - 166541628793765363764234332776 X[1] X[4] X[5] 2 2 + 82605368323414612131968036244 X[1] X[4] X[5] 3 - 15901107371894415317534999016 X[1] X[4] X[5] 4 + 801323924519912035913328820 X[1] X[5] 5 4 + 28654031183272935612808800 X[2] + 14312965969419644665319680 X[2] X[3] 4 - 238242197226873579788387520 X[2] X[4] 4 + 81824939190295593802853120 X[2] X[5] 3 2 - 41706405108492456974898528 X[2] X[3] 3 + 801759076265581346398142784 X[2] X[3] X[4] 3 - 342480709628285526743109664 X[2] X[3] X[5] 3 2 - 4277218081468863137148735072 X[2] X[4] 3 + 3983746566369230201954821776 X[2] X[4] X[5] 3 2 - 916870547936271692785941008 X[2] X[5] 2 3 - 17763478716258346203623552 X[2] X[3] 2 2 + 110918689854977517756166864 X[2] X[3] X[4] 2 2 - 37128206596485159127693872 X[2] X[3] X[5] 2 2 - 2336781350149778255627648432 X[2] X[3] X[4] 2 + 1811092864161343213708084096 X[2] X[3] X[4] X[5] 2 2 - 360485988494769238360657680 X[2] X[3] X[5] 2 3 + 30377032011223584560412655692 X[2] X[4] 2 2 - 39079221373834769568048547780 X[2] X[4] X[5] 2 2 + 16765048298668148244503988340 X[2] X[4] X[5] 2 3 - 2399298384533653629449943196 X[2] X[5] 4 + 29952513050566381230199680 X[2] X[3] 3 - 934576708272518023234915088 X[2] X[3] X[4] 3 + 419731422329214685272316752 X[2] X[3] X[5] 2 2 + 11660736268284398135194924776 X[2] X[3] X[4] 2 - 10265063013861252321897464272 X[2] X[3] X[4] X[5] 2 2 + 2261431151284383028516570952 X[2] X[3] X[5] 3 - 71506634652348943907689356848 X[2] X[3] X[4] 2 + 93063167331963613965260701456 X[2] X[3] X[4] X[5] 2 - 40364727369298794948923323296 X[2] X[3] X[4] X[5] 3 + 5838044519368030945750820480 X[2] X[3] X[5] 4 + 109110217175071627464241004028 X[2] X[4] 3 - 189928768268066457091615747072 X[2] X[4] X[5] 2 2 + 124147413378192911030548404200 X[2] X[4] X[5] 3 - 36126159653535602990943299664 X[2] X[4] X[5] 4 + 3949573760086097634381737036 X[2] X[5] 5 4 - 6910100282515062458778976 X[3] + 356971873130842162264931792 X[3] X[4] 4 - 164243250774914833016230160 X[3] X[5] 3 2 - 5524769551340760526295263968 X[3] X[4] 3 + 4964354523065946864805857072 X[3] X[4] X[5] 3 2 - 1114331183489012564546037040 X[3] X[5] 2 3 + 38098471835264821752977118348 X[3] X[4] 2 2 - 50192851653547486474964039244 X[3] X[4] X[5] 2 2 + 22009457778964531063153137924 X[3] X[4] X[5] 2 3 - 3212666977811341651215889476 X[3] X[5] 4 - 95720880999295412451804535166 X[3] X[4] 3 + 164276466470806444768490936068 X[3] X[4] X[5] 2 2 - 105326382841390713640017982184 X[3] X[4] X[5] 3 + 29891457956243938470346322316 X[3] X[4] X[5] 4 - 3167104504233050972465706858 X[3] X[5] 5 + 27756640453631269724082835929 X[4] 4 - 43742110498203794861119484191 X[4] X[5] 3 2 + 22447019974884702990832155902 X[4] X[5] 2 3 - 2486499175139372091698158738 X[4] X[5] 4 - 1146598366444406919079783319 X[4] X[5] 5 + 255553486110743507955829585 X[5] + 129734810581030541306801462368 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j -10 t - 6 t - 18 t + 20 t - 12 ) a[1, j] t = --------------------------------- / 5 4 3 2 ----- -t - 10 t + 18 t - 7 t + 1 j = 0 infinity ----- 4 3 2 \ j 4 t + 12 t + 5 t + 8 ) a[2, j] t = ------------------------------ / 5 4 3 2 ----- -t - 10 t + 18 t - 7 t + 1 j = 0 infinity ----- 4 3 2 \ j 13 t - 13 t + 3 t + 11 t + 7 ) a[3, j] t = ------------------------------- / 5 4 3 2 ----- -t - 10 t + 18 t - 7 t + 1 j = 0 infinity ----- 4 3 2 \ j -t + 7 t - 2 t - 6 t + 15 ) a[4, j] t = ------------------------------ / 5 4 3 2 ----- -t - 10 t + 18 t - 7 t + 1 j = 0 infinity ----- 4 3 2 \ j -19 t - t - 12 t + 6 t + 17 ) a[5, j] t = ------------------------------ / 5 4 3 2 ----- -t - 10 t + 18 t - 7 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 15 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 18208358111279230076272994528 X[1] 4 + 126355363660070295014043203200 X[1] X[2] 4 + 35693265926664255626712291488 X[1] X[3] 4 + 19106224000971404119281108624 X[1] X[4] 4 + 273489696400810607489783714416 X[1] X[5] 3 2 + 309167208250429844199123210240 X[1] X[2] 3 + 52371107305771481312728824320 X[1] X[2] X[3] 3 + 43649316914887148121893362944 X[1] X[2] X[4] 3 + 801520666110530910413501404416 X[1] X[2] X[5] 3 2 - 128301189564424844210072549696 X[1] X[3] 3 - 112086071697450970556006252352 X[1] X[3] X[4] 3 - 829374006005344363179919411392 X[1] X[3] X[5] 3 2 - 51508603189348378320048494800 X[1] X[4] 3 + 52749974544293430355021882784 X[1] X[4] X[5] 3 2 - 2636964889031495173746639456976 X[1] X[5] 2 3 + 329322022247392927662856529920 X[1] X[2] 2 2 - 171876954412327784278215493632 X[1] X[2] X[3] 2 2 - 90247975425740295212704317440 X[1] X[2] X[4] 2 2 + 554118131546531973400889604096 X[1] X[2] X[5] 2 2 - 594427050733904952076357424384 X[1] X[2] X[3] 2 - 751091688688988615391764750592 X[1] X[2] X[3] X[4] 2 - 3001030966659009969415560492288 X[1] X[2] X[3] X[5] 2 2 - 309344606915415741094135908672 X[1] X[2] X[4] 2 - 862754496444877211621571980416 X[1] X[2] X[4] X[5] 2 2 - 7052199839620863584023092597312 X[1] X[2] X[5] 2 3 - 101957344344829898818034061504 X[1] X[3] 2 2 - 310109828389935062283666740384 X[1] X[3] X[4] 2 2 - 641740201824912354529131270496 X[1] X[3] X[5] 2 2 - 214472443959397649563182375824 X[1] X[3] X[4] 2 - 2423511293582084802785529886432 X[1] X[3] X[4] X[5] 2 2 + 3525985043606920720298192143472 X[1] X[3] X[5] 2 3 - 41315080369909707114586801336 X[1] X[4] 2 2 - 1081623263747872693697289506136 X[1] X[4] X[5] 2 2 - 882292466446285350764146836008 X[1] X[4] X[5] 2 3 + 7846547590381703110953354675256 X[1] X[5] 4 + 149696193736096315798310641664 X[1] X[2] 3 - 329630496289874555235042734080 X[1] X[2] X[3] 3 - 221759743912967488484683024384 X[1] X[2] X[4] 3 - 93410604155293836368797136896 X[1] X[2] X[5] 2 2 - 677538033513261728901339611136 X[1] X[2] X[3] 2 - 1027846388687985041164679899136 X[1] X[2] X[3] X[4] 2 - 2983348632334617773318097283072 X[1] X[2] X[3] X[5] 2 2 - 436912075930870603083518704640 X[1] X[2] X[4] 2 - 1532187857589627695771421716480 X[1] X[2] X[4] X[5] 2 2 - 5328807882545583237706156209152 X[1] X[2] X[5] 3 + 96631676482792478128686998016 X[1] X[2] X[3] 2 - 100628971772120244939517867264 X[1] X[2] X[3] X[4] 2 + 329324009904076977665930701568 X[1] X[2] X[3] X[5] 2 - 157253380802430022204032307840 X[1] X[2] X[3] X[4] - 2819382237277664402533587627264 X[1] X[2] X[3] X[4] X[5] 2 + 9988998272871487929792211283840 X[1] X[2] X[3] X[5] 3 - 29492472984784702714551493568 X[1] X[2] X[4] 2 - 1877005535552965205467373611840 X[1] X[2] X[4] X[5] 2 + 2292817668974075323199518382528 X[1] X[2] X[4] X[5] 3 + 14659453882919524409819338020160 X[1] X[2] X[5] 4 + 237446100028510292759949008480 X[1] X[3] 3 + 510702384120903724256820255936 X[1] X[3] X[4] 3 + 2276726949589633370123399775040 X[1] X[3] X[5] 2 2 + 378144180780946461901846612752 X[1] X[3] X[4] 2 + 4279706392961623178002275901920 X[1] X[3] X[4] X[5] 2 2 + 4401675748827708208499390953744 X[1] X[3] X[5] 3 + 143219009823267109286701552816 X[1] X[3] X[4] 2 + 1825441908462180354819245856240 X[1] X[3] X[4] X[5] 2 + 10845423271920341981141036309520 X[1] X[3] X[4] X[5] 3 - 5314180044416451749225151415984 X[1] X[3] X[5] 4 + 34911476872866962906008203078 X[1] X[4] 3 - 31093755719017677326759133464 X[1] X[4] X[5] 2 2 + 4673361182489693507109331642660 X[1] X[4] X[5] 3 + 2333834294224379149769111835624 X[1] X[4] X[5] 4 - 9945728789983193791973779614522 X[1] X[5] 5 + 24658041689720253690151927808 X[2] 4 - 103667288566982310400291397632 X[2] X[3] 4 - 74002273522054650926106619904 X[2] X[4] 4 - 85482049483978122475462676480 X[2] X[5] 3 2 - 60532709513439336615673870336 X[2] X[3] 3 - 116024685931609116436616103936 X[2] X[3] X[4] 3 - 462935480443917483130779580416 X[2] X[3] X[5] 3 2 - 61981990313388384863736698368 X[2] X[4] 3 - 243005949816397317452700865536 X[2] X[4] X[5] 3 2 - 1011435699600054952670139946496 X[2] X[5] 2 3 + 454155930365900386201950566400 X[2] X[3] 2 2 + 965159181461684147342452422656 X[2] X[3] X[4] 2 2 + 1951218296284930172428800335872 X[2] X[3] X[5] 2 2 + 732828443950430014228121761280 X[2] X[3] X[4] 2 + 1953941661305763283715915533312 X[2] X[3] X[4] X[5] 2 2 + 5981748739591861665141469879808 X[2] X[3] X[5] 2 3 + 197246069655621115916272948224 X[2] X[4] 2 2 + 476245358726097952390426622464 X[2] X[4] X[5] 2 2 + 3023521633508775866406070706176 X[2] X[4] X[5] 2 3 + 6017067183177101511976500449792 X[2] X[5] 4 + 384427823741242635916985680512 X[2] X[3] 3 + 1287294576558664124136891562240 X[2] X[3] X[4] 3 + 2858649810531296835431806727424 X[2] X[3] X[5] 2 2 + 1543979241548422379573272503232 X[2] X[3] X[4] 2 + 8000427236274767640962938190208 X[2] X[3] X[4] X[5] 2 2 + 2119074231252807422423225088704 X[2] X[3] X[5] 3 + 809069028010607250150779512384 X[2] X[3] X[4] 2 + 6689866113074999907874377046976 X[2] X[3] X[4] X[5] 2 + 8790303694677195470299895008704 X[2] X[3] X[4] X[5] 3 - 8441415853758572694347827326912 X[2] X[3] X[5] 4 + 159717225525771929964697117992 X[2] X[4] 3 + 1742006396793112316470417365024 X[2] X[4] X[5] 2 2 + 4901240949738681438806502145712 X[2] X[4] X[5] 3 - 1472971641743195545350074313312 X[2] X[4] X[5] 4 - 9533363921842685128664962126232 X[2] X[5] 5 - 12000288325297746962888594912 X[3] 4 + 204167381579408353300770508944 X[3] X[4] 4 - 573800429642039685419869936528 X[3] X[5] 3 2 + 592395150244600374850404347216 X[3] X[4] 3 + 558369024695562520896802588512 X[3] X[4] X[5] 3 2 - 4205082218909023949037960906416 X[3] X[5] 2 3 + 566131992476145248658351123912 X[3] X[4] 2 2 + 3487396171773415824795280505896 X[3] X[4] X[5] 2 2 - 7222020241101834663844272553128 X[3] X[4] X[5] 2 3 - 5214267409127603435108167802184 X[3] X[5] 4 + 212646881146440376187935934698 X[3] X[4] 3 + 3256291495873804006985934759640 X[3] X[4] X[5] 2 2 - 2569127002339974437169779965828 X[3] X[4] X[5] 3 - 11141967490585783831943166667048 X[3] X[4] X[5] 4 + 2588872022498237013349941426410 X[3] X[5] 5 + 23422994540057612976883645685 X[4] 4 + 898615753617901917216718185479 X[4] X[5] 3 2 + 320322870140024678432870856722 X[4] X[5] 2 3 - 4627272285509596808872220925426 X[4] X[5] 4 - 1859031054195493926954077900087 X[4] X[5] 5 + 4642306874408575191470066389531 X[5] + 18244342051290722626068345880576 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j -19 t + 16 t - 11 t + 18 ) a[1, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 7 t + 17 t + 15 t - 2 t + 1 j = 0 infinity ----- 4 3 2 \ j 12 t + 9 t + t - 8 t - 18 ) a[2, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 7 t + 17 t + 15 t - 2 t + 1 j = 0 infinity ----- 4 3 2 \ j -17 t - 2 t + 9 t - 11 t - 10 ) a[3, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 7 t + 17 t + 15 t - 2 t + 1 j = 0 infinity ----- 4 3 2 \ j 6 t + 13 t + 12 t + 18 t + 18 ) a[4, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 7 t + 17 t + 15 t - 2 t + 1 j = 0 infinity ----- 4 3 2 \ j 2 t + t - 12 t + 4 t + 2 ) a[5, j] t = ------------------------------------ / 5 4 3 2 ----- -t - 7 t + 17 t + 15 t - 2 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 16 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 267182297115461242450739200000 X[1] 4 + 2855401695111840964608000000 X[1] X[2] 4 - 365823153474490736707993600000 X[1] X[3] 4 + 1098758288304050586137395200000 X[1] X[4] 4 - 598780871501792244497203200000 X[1] X[5] 3 2 - 3295743653072698371072000000 X[1] X[2] 3 - 6092029802470421006131200000 X[1] X[2] X[3] 3 + 27834833396883406255718400000 X[1] X[2] X[4] 3 + 315731534793306753945600000 X[1] X[2] X[5] 3 2 + 174214888879458802392893440000 X[1] X[3] 3 - 1171889195407543186467471360000 X[1] X[3] X[4] 3 + 663068216987484533722306560000 X[1] X[3] X[5] 3 2 + 1796190117135717722467368960000 X[1] X[4] 3 - 1975586560210909515594424320000 X[1] X[4] X[5] 3 2 + 530858127763687004303823360000 X[1] X[5] 2 3 - 2069549736505152996096000000 X[1] X[2] 2 2 - 1955068861661597392358400000 X[1] X[2] X[3] 2 2 - 8008082371392168551731200000 X[1] X[2] X[4] 2 2 + 204380199506026502899200000 X[1] X[2] X[5] 2 2 - 2452081749746590707632640000 X[1] X[2] X[3] 2 - 27821191830856470224547840000 X[1] X[2] X[3] X[4] 2 - 3005960037287777767503360000 X[1] X[2] X[3] X[5] 2 2 + 58541941391181138834186240000 X[1] X[2] X[4] 2 - 20628419015708016916654080000 X[1] X[2] X[4] X[5] 2 2 - 6144854226731558210540160000 X[1] X[2] X[5] 2 3 - 33615444665647745952968960000 X[1] X[3] 2 2 + 407992709840432979735014400000 X[1] X[3] X[4] 2 2 - 237919115993914989596626560000 X[1] X[3] X[5] 2 2 - 1399870375129498974230753280000 X[1] X[3] X[4] 2 + 1595180984045073082276377600000 X[1] X[3] X[4] X[5] 2 2 - 449727933298555634382486720000 X[1] X[3] X[5] 2 3 + 1459682543142478840882544640000 X[1] X[4] 2 2 - 2426409810691455500948367360000 X[1] X[4] X[5] 2 2 + 1321142072813118226658937600000 X[1] X[4] X[5] 2 3 - 232374560152950240848678880000 X[1] X[5] 4 + 173660797214259022656000000 X[1] X[2] 3 + 1435260957442631195443200000 X[1] X[2] X[3] 3 - 3923459671473138437222400000 X[1] X[2] X[4] 3 + 1611987368989700971238400000 X[1] X[2] X[5] 2 2 + 2008476852096763737550080000 X[1] X[2] X[3] 2 - 4332890385212381265899520000 X[1] X[2] X[3] X[4] 2 + 3050990797452500299633920000 X[1] X[2] X[3] X[5] 2 2 - 6178146104571815822561280000 X[1] X[2] X[4] 2 - 152396033919254905098240000 X[1] X[2] X[4] X[5] 2 2 + 1106008477637551178195520000 X[1] X[2] X[5] 3 + 2942824548583789015779840000 X[1] X[2] X[3] 2 - 3393072824744801261952000000 X[1] X[2] X[3] X[4] 2 + 6037830961840965563504640000 X[1] X[2] X[3] X[5] 2 - 32743114010220138007265280000 X[1] X[2] X[3] X[4] + 4558905987115611884774400000 X[1] X[2] X[3] X[4] X[5] 2 + 6199959399869663006820480000 X[1] X[2] X[3] X[5] 3 + 45778555323753948907683840000 X[1] X[2] X[4] 2 - 36152127152709259528949760000 X[1] X[2] X[4] X[5] 2 - 2399105217974683776019200000 X[1] X[2] X[4] X[5] 3 + 3715110163173856019541120000 X[1] X[2] X[5] 4 + 1978566340305563243550060800 X[1] X[3] 3 - 52057317551683109619293798400 X[1] X[3] X[4] 3 + 29923341820343226730343078400 X[1] X[3] X[5] 2 2 + 317599911406020344539191091200 X[1] X[3] X[4] 2 - 372294024260660056723635302400 X[1] X[3] X[4] X[5] 2 2 + 107969012196977702285176291200 X[1] X[3] X[5] 3 - 739389034469634100354488729600 X[1] X[3] X[4] 2 + 1271725598585441884710498508800 X[1] X[3] X[4] X[5] 2 - 723230893931465917174918348800 X[1] X[3] X[4] X[5] 3 + 134964733681796345532651369600 X[1] X[3] X[5] 4 + 589962892607435786812386508800 X[1] X[4] 3 - 1315537906644603824425456435200 X[1] X[4] X[5] 2 2 + 1086422033054824236588736972800 X[1] X[4] X[5] 3 - 389027900683939264006340275200 X[1] X[4] X[5] 4 + 50019160662526627498365478800 X[1] X[5] 5 4 - 2912855357482659734400000 X[2] - 71151113690184370305600000 X[2] X[3] 4 + 170503576748019530899200000 X[2] X[4] 4 - 56833366233755607487200000 X[2] X[5] 3 2 - 93620806215449963325120000 X[2] X[3] 3 + 1221318453507185388833280000 X[2] X[3] X[4] 3 - 641618252587399159298880000 X[2] X[3] X[5] 3 2 - 1830498154073750039086080000 X[2] X[4] 3 + 1564731966885826105071360000 X[2] X[4] X[5] 3 2 - 292602977062058715119280000 X[2] X[5] 2 3 + 78869634261432050135520000 X[2] X[3] 2 2 + 1440635667349906298471040000 X[2] X[3] X[4] 2 2 - 845696122155174954160080000 X[2] X[3] X[5] 2 2 - 2176497027282054219379200000 X[2] X[3] X[4] 2 + 2900841171394375437068160000 X[2] X[3] X[4] X[5] 2 2 - 946009540009400326801560000 X[2] X[3] X[5] 2 3 - 1497584081092233709393920000 X[2] X[4] 2 2 - 349550341671465664761600000 X[2] X[4] X[5] 2 2 + 1118344265081664726368160000 X[2] X[4] X[5] 2 3 - 256281588293539693126140000 X[2] X[5] 4 - 178995795823729360159281600 X[2] X[3] 3 + 2305483487314953276261196800 X[2] X[3] X[4] 3 - 1350402085579457295721756800 X[2] X[3] X[5] 2 2 - 1089661483944264438698342400 X[2] X[3] X[4] 2 + 4578109094543034525402124800 X[2] X[3] X[4] X[5] 2 2 - 2073386707398787121410442400 X[2] X[3] X[5] 3 - 11504836114530399898571980800 X[2] X[3] X[4] 2 + 5622111382511770183942502400 X[2] X[3] X[4] X[5] 2 + 3369383434644776884249977600 X[2] X[3] X[4] X[5] 3 - 1601454860677032636638599200 X[2] X[3] X[5] 4 + 12237008420127801354268262400 X[2] X[4] 3 - 15441610121042345456215449600 X[2] X[4] X[5] 2 2 + 2620007940172026431335334400 X[2] X[4] X[5] 3 + 2247878542018932839631830400 X[2] X[4] X[5] 4 - 589873425587873254352915100 X[2] X[5] 5 + 132965314758678048683195872 X[3] 4 + 1581045893571852137331665280 X[3] X[4] 4 - 567236559184132284838549680 X[3] X[5] 3 2 - 20194727194507753868402949120 X[3] X[4] 3 + 23178716266737022178240989440 X[3] X[4] X[5] 3 2 - 6461110902140789486227690320 X[3] X[5] 2 3 + 82168467999675764281188618240 X[3] X[4] 2 2 - 145347194594312868956916856320 X[3] X[4] X[5] 2 2 + 84553277232596634617373817920 X[3] X[4] X[5] 2 3 - 16253155084956458757086634840 X[3] X[5] 4 - 145752546504731105622616842240 X[3] X[4] 3 + 336140190764833974427854581760 X[3] X[4] X[5] 2 2 - 289038839528145825355470531840 X[3] X[4] X[5] 3 + 108873808318061685889367607360 X[3] X[4] X[5] 4 - 15108829510340197796036821290 X[3] X[5] 5 + 94920396857829686421145092096 X[4] 4 - 265802271939825631077198458880 X[4] X[5] 3 2 + 295439771068362204610597954560 X[4] X[5] 2 3 - 161113262519098189395533151360 X[4] X[5] 4 + 42408902087952892270109389080 X[4] X[5] 5 - 4211810622480121588150409871 X[5] + 148411502624003143680000000000 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j 19 t - 2 t - 5 t + 19 t - 11 ) a[1, j] t = --------------------------------- / 5 4 3 2 ----- -t - 2 t - t - 8 t + 12 t + 1 j = 0 infinity ----- 4 3 2 \ j -10 t - 14 t + 6 t - 7 t + 19 ) a[2, j] t = --------------------------------- / 5 4 3 2 ----- -t - 2 t - t - 8 t + 12 t + 1 j = 0 infinity ----- 4 3 2 \ j -4 t + 2 t + 8 t - 7 t - 16 ) a[3, j] t = --------------------------------- / 5 4 3 2 ----- -t - 2 t - t - 8 t + 12 t + 1 j = 0 infinity ----- 4 3 2 \ j -16 t + 5 t + 15 t - 15 t ) a[4, j] t = --------------------------------- / 5 4 3 2 ----- -t - 2 t - t - 8 t + 12 t + 1 j = 0 infinity ----- 4 3 2 \ j 16 t + 4 t + 16 t + 6 t - 12 ) a[5, j] t = --------------------------------- / 5 4 3 2 ----- -t - 2 t - t - 8 t + 12 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 17 Let 2 P(X[1], X[2], X[3], X[4], X[5]) = -1059921725625 X[1] - 1110775113000 X[1] X[2] + 1154840842050 X[1] X[3] 2 + 249254179650 X[1] X[4] + 2673326386500 X[1] X[5] - 291017091600 X[2] + 605124150120 X[2] X[3] + 130606502760 X[2] X[4] 2 + 1400794203600 X[2] X[5] - 314565061321 X[3] - 135787813266 X[3] X[4] 2 - 1456365324260 X[3] X[5] - 14653828809 X[4] - 314333482980 X[4] X[5] 2 - 1685660788900 X[5] + 1009968400 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j 8 t - 11 t - 10 t + 20 ) a[1, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 6 t - 4 t + 2 t + 14 t + 1 j = 0 infinity ----- 4 3 2 \ j t + 4 t - 10 t - 14 t + 7 ) a[2, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 6 t - 4 t + 2 t + 14 t + 1 j = 0 infinity ----- 4 3 2 \ j -18 t + 14 t + 10 t + 12 ) a[3, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 6 t - 4 t + 2 t + 14 t + 1 j = 0 infinity ----- 4 3 2 \ j 6 t - 3 t + 20 t + 20 t + 6 ) a[4, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 6 t - 4 t + 2 t + 14 t + 1 j = 0 infinity ----- 4 3 2 \ j 14 t - 13 t - 18 t - 8 t + 13 ) a[5, j] t = ----------------------------------- / 5 4 3 2 ----- -t + 6 t - 4 t + 2 t + 14 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 18 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 38413411504190672839349826432 X[1] 4 - 271407798021897090076825433632 X[1] X[2] 4 + 248965299514989342286153111184 X[1] X[3] 4 - 234341412678722387943066181488 X[1] X[4] 4 + 191278665675055834999893851328 X[1] X[5] 3 2 + 424613022219251957283558676736 X[1] X[2] 3 - 899943668590345266064420502464 X[1] X[2] X[3] 3 + 1221852923722566285212089957120 X[1] X[2] X[4] 3 - 798350602562321640922282233792 X[1] X[2] X[5] 3 2 + 464769630954275818741100955552 X[1] X[3] 3 - 1136393833508069844861202511648 X[1] X[3] X[4] 3 + 781782796439863933880135575392 X[1] X[3] X[5] 3 2 + 555348727818098734494186262848 X[1] X[4] 3 - 894188670355936203153619838496 X[1] X[4] X[5] 3 2 + 318475463203696476930939730560 X[1] X[5] 2 3 + 997351527590198328356926310080 X[1] X[2] 2 2 - 2009917849051498747945502477152 X[1] X[2] X[3] 2 2 - 861178372240027314145967434656 X[1] X[2] X[4] 2 2 - 610862611436936970258141568960 X[1] X[2] X[5] 2 2 + 1220573814702062250694765936784 X[1] X[2] X[3] 2 + 2141996299891191985187748145632 X[1] X[2] X[3] X[4] 2 + 388689823182220818978451297600 X[1] X[2] X[3] X[5] 2 2 - 1956410487551244396918467218992 X[1] X[2] X[4] 2 + 2255905353826811004681994461440 X[1] X[2] X[4] X[5] 2 2 - 273164352399052219846723082368 X[1] X[2] X[5] 2 3 - 196033192670746002270679722440 X[1] X[3] 2 2 - 1219350846045991917363926714280 X[1] X[3] X[4] 2 2 + 136134321477935320205728758224 X[1] X[3] X[5] 2 2 + 1855708281962324972679406962120 X[1] X[3] X[4] 2 - 2317658858017391518704713034304 X[1] X[3] X[4] X[5] 2 2 + 424421816294722008036851360576 X[1] X[3] X[5] 2 3 - 635990425808781952039709631576 X[1] X[4] 2 2 + 1504384946945417357721854182128 X[1] X[4] X[5] 2 2 - 1013898803381609126119837479744 X[1] X[4] X[5] 2 3 + 169169514573235428585745862400 X[1] X[5] 4 - 3537882888735799203584814190976 X[1] X[2] 3 + 11534996211200305073824794445248 X[1] X[2] X[3] 3 - 3941991032516146815336693301760 X[1] X[2] X[4] 3 + 6253992304541345168369752911808 X[1] X[2] X[5] 2 2 - 14035767683706110244352734422112 X[1] X[2] X[3] 2 + 8772892672397023805228531715040 X[1] X[2] X[3] X[4] 2 - 14935402049660751264865190746656 X[1] X[2] X[3] X[5] 2 2 + 96854582820445001887332825920 X[1] X[2] X[4] 2 + 3821657879295646784366183494880 X[1] X[2] X[4] X[5] 2 2 - 3824415324224978471645204820992 X[1] X[2] X[5] 3 + 7539023743581100345741595018672 X[1] X[2] X[3] 2 - 6294887003791251588453628465216 X[1] X[2] X[3] X[4] 2 + 11769019011819323694631117026768 X[1] X[2] X[3] X[5] 2 - 1000820987562898707541433890320 X[1] X[2] X[3] X[4] - 4914195544718190390104852263136 X[1] X[2] X[3] X[4] X[5] 2 + 5840856450002697101332016457536 X[1] X[2] X[3] X[5] 3 + 1306802010877294261532913090784 X[1] X[2] X[4] 2 - 1775969374514645871269374212272 X[1] X[2] X[4] X[5] 2 - 504622893590343886431770420096 X[1] X[2] X[4] X[5] 3 + 889052783267838156303245072064 X[1] X[2] X[5] 4 - 1504601302860599684948287410272 X[1] X[3] 3 + 1428929768976265170255072346616 X[1] X[3] X[4] 3 - 3046601488445081117924681180696 X[1] X[3] X[5] 2 2 + 803376869724334430460782527912 X[1] X[3] X[4] 2 + 1363130485551368409880096652600 X[1] X[3] X[4] X[5] 2 2 - 2176921469008852451613308243360 X[1] X[3] X[5] 3 - 1274766788214372179911318124696 X[1] X[3] X[4] 2 + 2012793897399269402928221657272 X[1] X[3] X[4] X[5] 2 - 49167778893626700773949521888 X[1] X[3] X[4] X[5] 3 - 618080317039947458129397788640 X[1] X[3] X[5] 4 + 350513482653445857201730090008 X[1] X[4] 3 - 1073091424818587234466992592024 X[1] X[4] X[5] 2 2 + 993608996997831118007104050624 X[1] X[4] X[5] 3 - 222523436286931658958818623584 X[1] X[4] X[5] 4 - 48513468223732843702291447296 X[1] X[5] 5 + 2794641689523622517243087737952 X[2] 4 - 11892802398544972689773198814512 X[2] X[3] 4 + 5340226370239609969682064165392 X[2] X[4] 4 - 6793762198390423000892573043968 X[2] X[5] 3 2 + 20255110218282463234150539566384 X[2] X[3] 3 - 17601344466654182660943506732704 X[2] X[3] X[4] 3 + 22947679167528023596177089488256 X[2] X[3] X[5] 3 2 + 3278326001761215577481474357424 X[2] X[4] 3 - 9833423238224217780757987316672 X[2] X[4] X[5] 3 2 + 6442338950694882330755783527936 X[2] X[5] 2 3 - 17245828975908365492556029257432 X[2] X[3] 2 2 + 21686402611781894572946477998488 X[2] X[3] X[4] 2 2 - 29067790511772658560942763708480 X[2] X[3] X[5] 2 2 - 7624239489743032991093290364808 X[2] X[3] X[4] 2 + 23908289947493761031314337211360 X[2] X[3] X[4] X[5] 2 2 - 16144841162268814263418169117632 X[2] X[3] X[5] 2 3 + 331588977475550279082651356936 X[2] X[4] 2 2 - 3736464399808369814656164133088 X[2] X[4] X[5] 2 2 + 6416257991801655361277583328128 X[2] X[4] X[5] 2 3 - 2955040431152150785712558660928 X[2] X[5] 4 + 7335131601688303755461558051790 X[2] X[3] 3 - 11824747667092015062677463062856 X[2] X[3] X[4] 3 + 16350320985984110650747040416064 X[2] X[3] X[5] 2 2 + 5810615335851229119984240524340 X[2] X[3] X[4] 2 - 19257100411985756311405136681392 X[2] X[3] X[4] X[5] 2 2 + 13471749891362118962673951571968 X[2] X[3] X[5] 3 - 236809379563835770455334021768 X[2] X[3] X[4] 2 + 5423381604139868775828791339616 X[2] X[3] X[4] X[5] 2 - 10130450266333183778102098978112 X[2] X[3] X[4] X[5] 3 + 4862926646371473276179107162176 X[2] X[3] X[5] 4 - 307901669251621775489265464786 X[2] X[4] 3 + 317882578161414009468057686608 X[2] X[4] X[5] 2 2 + 1049039684651759439709723312192 X[2] X[4] X[5] 3 - 1690855657062822991721960875392 X[2] X[4] X[5] 4 + 646194654949372476620880173472 X[2] X[5] 5 - 1245838660062050359213254497831 X[3] 4 + 2404341505781034692900480069025 X[3] X[4] 4 - 3442398614493800582350607267248 X[3] X[5] 3 2 - 1441797137601747933700520250150 X[3] X[4] 3 + 5127306565635852840445104243720 X[3] X[4] X[5] 3 2 - 3738060532654387375669522272016 X[3] X[5] 2 3 - 47283795866691547790139547518 X[3] X[4] 2 2 - 1880221758977469262640639043400 X[3] X[4] X[5] 2 2 + 3948449575712549652829036338592 X[3] X[4] X[5] 2 3 - 1993871625264509524677141641040 X[3] X[5] 4 + 311597459703112425897837396157 X[3] X[4] 3 - 471697676169514084304214463976 X[3] X[4] X[5] 2 2 - 604427882506022409996425766224 X[3] X[4] X[5] 3 + 1271765847464046523932442538304 X[3] X[4] X[5] 4 - 519998636275479741186214856784 X[3] X[5] 5 - 74403057872111843137463697075 X[4] 4 + 274451319026327050825009549080 X[4] X[5] 3 2 - 294638163494300931650662717440 X[4] X[5] 2 3 + 7972161336947184962260260240 X[4] X[5] 4 + 136191078057459204872069171760 X[4] X[5] 5 - 52316664017289019800959309760 X[5] + 3432587949868763024302335258112 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j 19 t - 17 t - 11 t - 13 t + 1 ) a[1, j] t = ------------------------------------- / 5 4 3 2 ----- -t - 15 t - 14 t - 11 t - 4 t + 1 j = 0 infinity ----- 4 3 2 \ j -11 t - 13 t - 6 t - 11 t - 20 ) a[2, j] t = ------------------------------------- / 5 4 3 2 ----- -t - 15 t - 14 t - 11 t - 4 t + 1 j = 0 infinity ----- 4 3 2 \ j -5 t + 2 t + 18 t - 3 t - 12 ) a[3, j] t = ------------------------------------- / 5 4 3 2 ----- -t - 15 t - 14 t - 11 t - 4 t + 1 j = 0 infinity ----- 4 2 \ j 13 t + 10 t + 11 t + 2 ) a[4, j] t = ------------------------------------- / 5 4 3 2 ----- -t - 15 t - 14 t - 11 t - 4 t + 1 j = 0 infinity ----- 4 3 2 \ j -10 t - t - 15 t + 8 t - 16 ) a[5, j] t = ------------------------------------- / 5 4 3 2 ----- -t - 15 t - 14 t - 11 t - 4 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 19 Let P(X[1], X[2], X[3], X[4], X[5]) = -9817 X[1] + 1414 X[2] - 20427 X[3] - 40660 X[4] - 32052 X[5] + 17482 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j -3 t - 9 t + 19 t + 18 t + 5 ) a[1, j] t = ----------------------------------- / 5 4 3 2 ----- -t - t - 20 t + 10 t + 11 t + 1 j = 0 infinity ----- 4 3 2 \ j 4 t + 20 t + 18 t + 10 t - 3 ) a[2, j] t = ----------------------------------- / 5 4 3 2 ----- -t - t - 20 t + 10 t + 11 t + 1 j = 0 infinity ----- 4 3 2 \ j -13 t + 7 t - t - 6 t - 7 ) a[3, j] t = ----------------------------------- / 5 4 3 2 ----- -t - t - 20 t + 10 t + 11 t + 1 j = 0 infinity ----- 4 3 2 \ j -4 t + t + 6 t + 16 t + 5 ) a[4, j] t = ----------------------------------- / 5 4 3 2 ----- -t - t - 20 t + 10 t + 11 t + 1 j = 0 infinity ----- 4 3 \ j 15 t - t - 15 t - 3 ) a[5, j] t = ----------------------------------- / 5 4 3 2 ----- -t - t - 20 t + 10 t + 11 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- Theorem Number, 20 Let 5 P(X[1], X[2], X[3], X[4], X[5]) = 1475682372991127137792046153463296 X[1] 4 - 5235204753552438776091199042288640 X[1] X[2] 4 + 8325949976465443229801183779581440 X[1] X[3] 4 - 14973155710952195721198940230965760 X[1] X[4] 4 + 8371342410379133814239010960823040 X[1] X[5] 3 2 + 7369643072518203703167148766666240 X[1] X[2] 3 - 23594100971350419410899858868500480 X[1] X[2] X[3] 3 + 42400125726015834576808139261560320 X[1] X[2] X[4] 3 - 23634550833631733603698268727585280 X[1] X[2] X[5] 3 2 + 18679854272376323612299062418634240 X[1] X[3] 3 - 67213843173838388230038536201809920 X[1] X[3] X[4] 3 + 37681161106151634863424087384052480 X[1] X[3] X[5] 3 2 + 60696319060130516621588195602260480 X[1] X[4] 3 - 67979698873855984287094794186608640 X[1] X[4] X[5] 3 2 + 18956453679822689212360105630282880 X[1] X[5] 2 3 - 5142562266682707575052109723995520 X[1] X[2] 2 2 + 24845645511539957478197926807455360 X[1] X[2] X[3] 2 2 - 44656276092220598022725759867744640 X[1] X[2] X[4] 2 2 + 24825538247873411734436817273610560 X[1] X[2] X[5] 2 2 - 39599574642428785902810275965340160 X[1] X[2] X[3] 2 + 142502201689097249866976663810465280 X[1] X[2] X[3] X[4] 2 - 79654656479897802678025413657160320 X[1] X[2] X[3] X[5] 2 2 - 128574781750044535274970557333944320 X[1] X[2] X[4] 2 + 143593667047430868310626701376581760 X[1] X[2] X[4] X[5] 2 2 - 39927582339450266412312122642581920 X[1] X[2] X[5] 2 3 + 20839677062797519906611880057208320 X[1] X[3] 2 2 - 112481372028267675941499868570352640 X[1] X[3] X[4] 2 2 + 63205331820501498080780846556917760 X[1] X[3] X[5] 2 2 + 203224864413414187534001505616811520 X[1] X[3] X[4] 2 - 228242464499107385148947865007138560 X[1] X[3] X[4] X[5] 2 2 + 63827969529917807090569627505893920 X[1] X[3] X[5] 2 3 - 122871798424122347302368183845683200 X[1] X[4] 2 2 + 206738557660307658055869276215097600 X[1] X[4] X[5] 2 2 - 115480882357032210444258300229327200 X[1] X[4] X[5] 2 3 + 21411781197382960912639138026574000 X[1] X[5] 4 + 1777958338838549379279487483482880 X[1] X[2] 3 - 11515106183056264834363884779710720 X[1] X[2] X[3] 3 + 20717281015929402027341447624417280 X[1] X[2] X[4] 3 - 11492023267620397741317279672013120 X[1] X[2] X[5] 2 2 + 27695923987813726044415212933290880 X[1] X[2] X[3] 2 - 99767510706769962831741271755536640 X[1] X[2] X[3] X[4] 2 + 55626847080855199695461439570812160 X[1] X[2] X[3] X[5] 2 2 + 90027284681126205891384169704831360 X[1] X[2] X[4] 2 - 100289989431492989723447705746980480 X[1] X[2] X[4] X[5] 2 2 + 27816853712479231988196741995432160 X[1] X[2] X[5] 3 - 29339015040081340511289317056789120 X[1] X[2] X[3] 2 + 158537153173157599455768443997709440 X[1] X[2] X[3] X[4] 2 - 88839884448698755752004161314616960 X[1] X[2] X[3] X[5] 2 - 286435960950272984602949574760014720 X[1] X[2] X[3] X[4] + 320803965000777305714955410316188160 X[1] X[2] X[3] X[4] X[5] 2 - 89461631929587119809551526635261120 X[1] X[2] X[3] X[5] 3 + 173013598761172406594098445395670400 X[1] X[2] X[4] 2 - 290327817087285162383744123708016000 X[1] X[2] X[4] X[5] 2 + 161737015687028905789459110185308800 X[1] X[2] X[4] X[5] 3 - 29907714856519642405917037411586800 X[1] X[2] X[5] 4 + 11559435953112676753152337531886080 X[1] X[3] 3 - 83195852529000132742629672205678080 X[1] X[3] X[4] 3 + 46843586168334622706508441036569920 X[1] X[3] X[5] 2 2 + 225480467772522488632888946159055360 X[1] X[3] X[4] 2 - 253843013036053093608516315325423680 X[1] X[3] X[4] X[5] 2 2 + 71165453936341086021273878268900960 X[1] X[3] X[5] 3 - 272750865065837998263221274168076800 X[1] X[3] X[4] 2 + 460214498002322805665206971121790400 X[1] X[3] X[4] X[5] 2 - 257817856462995901049873714385316800 X[1] X[3] X[4] X[5] 3 + 47944213924831876805795240153488000 X[1] X[3] X[5] 4 + 124213608889799917169700590063293440 X[1] X[4] 3 - 279060664885292854618110545431400640 X[1] X[4] X[5] 2 2 + 234167514559476880913229917856025440 X[1] X[4] X[5] 3 - 86968858272042580326524867592025440 X[1] X[4] X[5] 4 + 12060346164943685767225135237830040 X[1] X[5] 5 - 243542269755959425682496605578528 X[2] 4 + 1980784712916230347290617669930240 X[2] X[3] 4 - 3570090246801493813989598515932160 X[2] X[4] 4 + 1977264326209939239985194927850640 X[2] X[5] 3 2 - 6386012448531639105161381337959680 X[2] X[3] 3 + 23047071168178061655615837559578240 X[2] X[3] X[4] 3 - 12826014651843957960705477451842560 X[2] X[3] X[5] 3 2 - 20818432484562115419942769952492160 X[2] X[4] 3 + 23145412649051474632534376789930880 X[2] X[4] X[5] 3 2 - 6406933302245589397277194993690960 X[2] X[5] 2 3 + 10207095007772619251542635203754880 X[2] X[3] 2 2 - 55268715350734693695497977512197760 X[2] X[3] X[4] 2 2 + 30903435869068919467330668538227840 X[2] X[3] X[5] 2 2 + 99957349514079035416583031965719680 X[2] X[3] X[4] 2 - 111690706207099912204445413178639040 X[2] X[3] X[4] X[5] 2 2 + 31074425272883423658569513453237280 X[2] X[3] X[5] 2 3 - 60381583762792601243111182268860800 X[2] X[4] 2 2 + 101088224251485101502213578841374400 X[2] X[4] X[5] 2 2 - 56184267042665270716579992388576800 X[2] X[4] X[5] 2 3 + 10365377209791189698944847101021000 X[2] X[5] 4 - 8093349008844217727516843292439040 X[2] X[3] 3 + 58386702085693197284980894907773440 X[2] X[3] X[4] 3 - 32795674234185926242839670648266560 X[2] X[3] X[5] 2 2 - 158411666335059213398025606528122880 X[2] X[3] X[4] 2 + 177877787378421705160750247350405440 X[2] X[3] X[4] X[5] 2 2 - 49738322696183764736559063737551680 X[2] X[3] X[5] 3 + 191603224882185832843477785904089600 X[2] X[3] X[4] 2 - 322455725045045374649845700039601600 X[2] X[3] X[4] X[5] 2 + 180172637987764014689453083753452000 X[2] X[3] X[4] X[5] 3 - 33417249945601271533128226561392800 X[2] X[3] X[5] 4 - 87163156033724345002995387065541120 X[2] X[4] 3 + 195339476687737232436314066271414720 X[2] X[4] X[5] 2 2 - 163507408308590336425257701718273120 X[2] X[4] X[5] 3 + 60574200819810404459203293097821120 X[2] X[4] X[5] 4 - 8379001789662325416057996403806170 X[2] X[5] 5 + 2548774451140656893279949214592128 X[3] 4 - 22942939836901153434414743222501760 X[3] X[4] 4 + 12942340848826654203112041874000640 X[3] X[5] 3 2 + 82920525258974764623792167196207360 X[3] X[4] 3 - 93547124403565431645010568804993280 X[3] X[4] X[5] 3 2 + 26285048059451803290920906357047360 X[3] X[5] 2 3 - 150466603954503945221512982630956800 X[3] X[4] 2 2 + 254507286413740809367194681215923200 X[3] X[4] X[5] 2 2 - 142948270919226511884029862305779200 X[3] X[4] X[5] 2 3 + 26654774217221544376120179377168800 X[3] X[5] 4 + 137093125703932257830898581335777920 X[3] X[4] 3 - 308883860292532370205845245132581120 X[3] X[4] X[5] 2 2 + 259967254074903742708359800332557120 X[3] X[4] X[5] 3 - 96845097982633845020928387560376720 X[3] X[4] X[5] 4 + 13470756879333028769300985765632120 X[3] X[5] 5 - 50161686322280351453971647361918848 X[4] 4 + 141058871988652795814650780863010560 X[4] X[5] 3 2 - 158047702807581514761904809163889280 X[4] X[5] 2 3 + 88178302690707490060490765713498320 X[4] X[5] 4 - 24493089175329983745184702532924040 X[4] X[5] 5 + 2709311475387603041591823649937077 X[5] + 3125218756125085750600251680700000 Define , 5, sequences a[i,j] i goes from 1 to, 5, and j from 0 to infinity in terms of the generating functions infinity ----- 4 3 2 \ j 11 t - 17 t + 3 t - 16 t - 19 ) a[1, j] t = ----------------------------------- / 5 4 3 2 ----- -t + t + 18 t - 18 t + 11 t + 1 j = 0 infinity ----- 4 3 2 \ j -19 t + 14 t + 19 t + 3 t - 19 ) a[2, j] t = ----------------------------------- / 5 4 3 2 ----- -t + t + 18 t - 18 t + 11 t + 1 j = 0 infinity ----- 4 3 2 \ j -12 t + 14 t - 7 t - 19 t - 5 ) a[3, j] t = ----------------------------------- / 5 4 3 2 ----- -t + t + 18 t - 18 t + 11 t + 1 j = 0 infinity ----- 4 3 2 \ j -6 t - 19 t - 2 t - 20 t - 12 ) a[4, j] t = ----------------------------------- / 5 4 3 2 ----- -t + t + 18 t - 18 t + 11 t + 1 j = 0 infinity ----- 4 3 2 \ j -20 t - 20 t + 2 t + 6 t - 12 ) a[5, j] t = ----------------------------------- / 5 4 3 2 ----- -t + t + 18 t - 18 t + 11 t + 1 j = 0 then for each j from 0 to infinity we have P(a[1, j], a[2, j], a[3, j], a[4, j], a[5, j]) = 0 ----------------------------- ---------------------------------------- This concludes this article that took, 25.245, seconds to generate