All the Primality Tests Inspired by an alphabet of 2 to , 3, letters and sets\ of forbidden words of size 1 where the words are constants and of lengt\ h from 2 to, 7 By Shalosh B. Ekhad [[{1, 2, 3}, {[1, 1], [2, 2, 2], [3, 3, 3, 3]}], (5*s^5+14*s^4+18*s^3+12*s^2-6) /(s+1)/(2*s^5+3*s^4+4*s^3+2*s^2+s-1), [6, 9]] [[{1, 2, 3}, {[1, 1], [2, 2, 2], [3, 3, 3, 3, 3]}], (5*s^6+14*s^5+21*s^4+24*s^3 +15*s^2-7)/(2*s^7+5*s^6+7*s^5+7*s^4+6*s^3+3*s^2-1), [6, 9, 25, 49, 125, 310, 318, 4873, 16854, 27759, 68910]] [[{1, 2, 3}, {[1, 1], [2, 2, 2], [3, 3, 3, 3, 3, 3]}], (5*s^7+14*s^6+21*s^5+28* s^4+30*s^3+18*s^2-8)/(s^3+2*s^2+2*s+1)/(2*s^5+s^4+s^3+s^2+2*s-1), [6, 9, 25, 66 , 125, 225, 1179, 23422, 55395]] [[{1, 2, 3}, {[1, 1], [2, 2, 2], [3, 3, 3, 3, 3, 3, 3]}], (5*s^8+14*s^7+21*s^6+ 28*s^5+35*s^4+36*s^3+21*s^2-9)/(2*s^9+5*s^8+7*s^7+7*s^6+7*s^5+7*s^4+6*s^3+3*s^2 -1), [6, 9, 25, 27, 30, 444, 1681, 6773, 7964, 12988, 17161]] [[{1, 2, 3}, {[1, 1], [2, 2, 2, 2], [3, 3, 3, 3, 3]}], (5*s^7+16*s^6+30*s^5+36* s^4+30*s^3+18*s^2-8)/(s+1)/(2*s^7+3*s^6+5*s^5+5*s^4+4*s^3+2*s^2+s-1), [6, 9, 12 , 27, 77, 169, 2532, 5889, 8149, 11823, 45106]] [[{1, 2, 3}, {[1, 1], [2, 2, 2, 2], [3, 3, 3, 3, 3, 3]}], (5*s^7+11*s^6+19*s^5+ 21*s^4+24*s^3+12*s^2+9*s-9)/(s+1)/(2*s^7+s^6+4*s^5+s^4+4*s^3+2*s-1), [6, 9, 12, 56, 588, 11118, 11480, 12191, 18582]] [[{1, 2, 3}, {[1, 1], [2, 2, 2, 2], [3, 3, 3, 3, 3, 3, 3]}], (5*s^9+16*s^8+30*s ^7+40*s^6+50*s^5+54*s^4+42*s^3+24*s^2-10)/(s+1)/(2*s^9+3*s^8+5*s^7+5*s^6+5*s^5+ 5*s^4+4*s^3+2*s^2+s-1), [6, 9, 12, 154, 252, 434, 494, 1479, 13459, 17557]] [[{1, 2, 3}, {[1, 1], [2, 2, 2, 2, 2], [3, 3, 3, 3, 3, 3]}], (5*s^9+16*s^8+33*s ^7+52*s^6+60*s^5+54*s^4+42*s^3+24*s^2-10)/(s+1)/(2*s^9+3*s^8+5*s^7+6*s^6+7*s^5+ 5*s^4+4*s^3+2*s^2+s-1), [6, 9, 25, 45, 125, 266, 312, 625, 1645, 1681, 2050, 3125, 5370, 18657, 30673, 70614]] [[{1, 2, 3}, {[1, 1], [2, 2, 2, 2, 2], [3, 3, 3, 3, 3, 3, 3]}], (5*s^10+16*s^9+ 33*s^8+52*s^7+65*s^6+72*s^5+63*s^4+48*s^3+27*s^2-11)/(2*s^11+5*s^10+8*s^9+11*s^ 8+13*s^7+13*s^6+12*s^5+9*s^4+6*s^3+3*s^2-1), [6, 9, 25, 3560, 3945, 6851, 11881 , 15495, 52180]] [[{1, 2, 3}, {[1, 1], [2, 2, 2, 2, 2, 2], [3, 3, 3, 3, 3, 3, 3]}], (5*s^11+16*s ^10+33*s^9+56*s^8+80*s^7+90*s^6+84*s^5+72*s^4+54*s^3+30*s^2-12)/(s+1)/(2*s^11+3 *s^10+5*s^9+6*s^8+8*s^7+8*s^6+7*s^5+5*s^4+4*s^3+2*s^2+s-1), [6, 9, 25, 670, 852 , 6625, 49932, 66254]] [[{1, 2, 3}, {[1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3, 3]}], (6*s^8+22*s^7+45*s^6+ 64*s^5+65*s^4+48*s^3+21*s^2-9)/(2*s^9+6*s^8+11*s^7+15*s^6+16*s^5+13*s^4+8*s^3+3 *s^2-1), [6, 25, 125, 196, 289, 2752, 9994, 17219, 18968]] [[{1, 2, 3}, {[1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3, 3, 3]}], (6*s^9+22*s^8+45*s ^7+68*s^6+80*s^5+78*s^4+56*s^3+24*s^2-10)/(s^3+2*s^2+2*s+1)/(2*s^7+2*s^6+3*s^5+ 3*s^4+3*s^3+s^2+2*s-1), [6, 20, 25, 40, 196, 306, 798, 7921, 37944, 81679]] [[{1, 2, 3}, {[1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3, 3, 3, 3]}], (6*s^10+22*s^9+ 45*s^8+68*s^7+85*s^6+96*s^5+91*s^4+64*s^3+27*s^2-11)/(2*s^11+6*s^10+11*s^9+15*s ^8+17*s^7+17*s^6+16*s^5+13*s^4+8*s^3+3*s^2-1), [6, 20, 25, 80, 289, 1687, 1990, 2990, 4125, 21867, 57011, 58581, 60451, 83355]] [[{1, 2, 3}, {[1, 1, 1], [2, 2, 2, 2, 2], [3, 3, 3, 3, 3, 3]}], (6*s^10+22*s^9+ 48*s^8+80*s^7+105*s^6+108*s^5+91*s^4+64*s^3+27*s^2-11)/(s^2+s+1)/(2*s^9+4*s^8+5 *s^7+7*s^6+8*s^5+6*s^4+4*s^3+3*s^2+s-1), [6, 35, 560, 2264, 2685, 3157, 8877, 18769]] [[{1, 2, 3}, {[1, 1, 1], [2, 2, 2, 2, 2], [3, 3, 3, 3, 3, 3, 3]}], (6*s^11+22*s ^10+48*s^9+80*s^8+110*s^7+126*s^6+126*s^5+104*s^4+72*s^3+30*s^2-12)/(2*s^12+6*s ^11+11*s^10+16*s^9+20*s^8+22*s^7+21*s^6+18*s^5+13*s^4+8*s^3+3*s^2-1), [6, 20, 49, 88, 532, 639, 2209, 2501, 3354, 6940, 27110, 50928]] [[{1, 2, 3}, {[1, 1, 1], [2, 2, 2, 2, 2, 2], [3, 3, 3, 3, 3, 3, 3]}], (6*s^12+ 22*s^11+48*s^10+84*s^9+125*s^8+156*s^7+161*s^6+144*s^5+117*s^4+80*s^3+33*s^2-13 )/(s^2+s+1)/(2*s^11+4*s^10+5*s^9+7*s^8+9*s^7+9*s^6+8*s^5+6*s^4+4*s^3+3*s^2+s-1) , [6, 20, 88, 100, 160, 176, 240, 352, 944, 1504, 3504, 3856, 3965, 10591, 60009]] [[{1, 2, 3}, {[1, 1, 1, 1], [2, 2, 2, 2, 2], [3, 3, 3, 3, 3, 3]}], (6*s^11+24*s ^10+57*s^9+100*s^8+140*s^7+162*s^6+154*s^5+120*s^4+72*s^3+30*s^2-12)/(s+1)/(2*s ^11+4*s^10+8*s^9+11*s^8+14*s^7+14*s^6+13*s^5+9*s^4+6*s^3+2*s^2+s-1), [6, 184, 703, 2418, 17149, 20538, 35927]] [[{1, 2, 3}, {[1, 1, 1, 1], [2, 2, 2, 2, 2], [3, 3, 3, 3, 3, 3, 3]}], (6*s^12+ 24*s^11+57*s^10+100*s^9+145*s^8+180*s^7+189*s^6+176*s^5+135*s^4+80*s^3+33*s^2-\ 13)/(2*s^13+6*s^12+12*s^11+19*s^10+25*s^9+29*s^8+30*s^7+27*s^6+22*s^5+15*s^4+8* s^3+3*s^2-1), [6, 121, 152, 240, 1331, 3363, 4152, 14641, 17161]] [[{1, 2, 3}, {[1, 1, 1, 1], [2, 2, 2, 2, 2, 2], [3, 3, 3, 3, 3, 3, 3]}], (6*s^ 13+24*s^12+57*s^11+104*s^10+160*s^9+210*s^8+238*s^7+232*s^6+198*s^5+150*s^4+88* s^3+36*s^2-14)/(s+1)/(2*s^13+4*s^12+8*s^11+11*s^10+15*s^9+17*s^8+18*s^7+16*s^6+ 13*s^5+9*s^4+6*s^3+2*s^2+s-1), [6, 30, 3464, 13482, 27227, 90029, 91549]] [[{1, 2, 3}, {[1, 1, 1, 1, 1], [2, 2, 2, 2, 2, 2], [3, 3, 3, 3, 3, 3, 3]}], (6* s^14+24*s^13+60*s^12+116*s^11+185*s^10+252*s^9+301*s^8+320*s^7+297*s^6+240*s^5+ 165*s^4+96*s^3+39*s^2-15)/(2*s^15+6*s^14+12*s^13+20*s^12+29*s^11+37*s^10+42*s^9 +43*s^8+40*s^7+33*s^6+24*s^5+15*s^4+8*s^3+3*s^2-1), [6, 361, 1369, 1681, 1855, 2321, 4557, 7270, 28467, 47082, 86870, 90802]] The set of patterns with, followed by the generating function, with the leas\ t number of pseudoprimes less than, 100000, is: [[{1, 2, 3}, {[1, 1], [2, 2, 2], [3, 3, 3, 3]}], (5*s^5+14*s^4+18*s^3+12*s^2-6) /(s+1)/(2*s^5+3*s^4+4*s^3+2*s^2+s-1), [6, 9]] --------------------- This took, 7878.774, seconds.