Here is a full dicovery and proof of the product of three Chebychev polynomi\ als analog of Louis Shapiro's generating function given in his paper A combinatorial proof of a Chebyshev polynomial identit\ y, Discrete Math. 34 (1981), pages 203-206. Theorem: Let F(n) be the sequence defined by the recurrence 2 2 F(n) = 4 a b F(n - 1) + (-4 a + 2 - 4 b ) F(n - 2) + 4 a b F(n - 3) - F(n - 4) subject to the initial conditions 2 2 F(0) = 1, F(1) = 4 a b, F(2) = (-1 + 4 a ) (-1 + 4 b ), 3 3 F(3) = (-4 a + 8 a ) (-4 b + 8 b ) Equivalently, the ordinary generating function of F(n) w.r.t. to t is infinity ----- 2 \ n 1 - t ) F(n) t = --------------------------------------------------- / 2 2 2 3 4 ----- 1 - 4 a b t - (-4 a + 2 - 4 b ) t - 4 a b t + t n = 0 Also Let G(n) be the sequence defined by the recurrence G(n) = 2 c G(n - 1) - G(n - 2) subject to the initial conditions G(0) = 1, G(1) = 2 c Equivalently, the ordinary generating function of G(n) w.r.t. to t is infinity ----- \ n 1 ) G(n) t = -------------- / 2 ----- 1 - 2 c t + t n = 0 Let H(n)=F(n)G(n), then infinity ----- \ n 2 2 2 2 3 ) H(n) t = (1 + (-4 a - 4 b - 4 c + 3) t + 16 a b c t / ----- n = 0 2 2 2 4 6 / + (-4 a - 4 b - 4 c + 3) t + t ) / (1 - 8 a b c t / 2 2 2 2 2 2 2 2 2 2 - (-16 a b + 8 a - 16 a c - 16 b c + 8 b - 4 + 8 c ) t 2 2 2 3 - 8 (4 a + 4 b - 5 + 4 c ) a b c t 4 2 2 2 2 4 2 4 2 4 - (-16 a - 64 a b c + 16 a - 16 b + 16 b - 6 - 16 c + 16 c ) t 2 2 2 5 - 8 (4 a + 4 b - 5 + 4 c ) a b c t 2 2 2 2 2 2 2 2 2 6 - (-16 a b + 8 a - 16 a c - 16 b c + 8 b - 4 + 8 c ) t 7 8 - 8 a b c t + t ) or equivalently, H(n) is the sequence defined by the recurrence H(n) = 8 a b c H(n - 1) 2 2 2 2 2 2 2 2 2 + (-16 a b + 8 a - 16 a c - 16 b c + 8 b - 4 + 8 c ) H(n - 2) 2 2 2 + 8 (4 a + 4 b - 5 + 4 c ) a b c H(n - 3) + 4 2 2 2 2 4 2 4 2 (-16 a - 64 a b c + 16 a - 16 b + 16 b - 6 - 16 c + 16 c ) H(n - 4) 2 2 2 + 8 (4 a + 4 b - 5 + 4 c ) a b c H(n - 5) 2 2 2 2 2 2 2 2 2 + (-16 a b + 8 a - 16 a c - 16 b c + 8 b - 4 + 8 c ) H(n - 6) + 8 a b c H(n - 7) - H(n - 8) subject to the initial conditions 2 2 2 H(0) = 1, H(1) = 8 a b c, H(2) = (-1 + 4 a ) (-1 + 4 b ) (-1 + 4 c ), 3 3 3 2 2 H(3) = (-4 a + 8 a ) (-4 b + 8 b ) (-4 c + 8 c ), H(4) = (144 a b 2 4 2 4 2 4 4 4 2 4 - 192 a b - 12 a - 192 a b + 256 a b + 16 a - 12 b + 16 b + 1) 2 4 3 3 3 5 3 (-12 c + 16 c + 1), H(5) = (1024 a b - 1024 a b - 192 a b 5 3 5 5 5 3 5 - 1024 a b + 1024 a b + 192 a b - 192 a b + 192 a b + 36 a b) 3 5 4 4 4 6 4 2 (-32 c + 32 c + 6 c), H(6) = (6400 a b - 5120 a b - 1920 a b 4 6 4 6 6 6 2 6 2 4 + 80 a - 5120 a b + 4096 a b + 1536 a b - 64 a - 1920 a b 2 6 2 2 2 4 6 2 + 1536 a b + 576 a b - 24 a + 80 b - 64 b - 24 b + 1) 4 6 2 5 5 5 7 (-80 c + 64 c + 24 c - 1), H(7) = (36864 a b - 24576 a b 5 3 5 7 5 7 7 7 3 - 15360 a b + 1536 a b - 24576 a b + 16384 a b + 10240 a b 7 3 5 3 7 3 3 3 - 1024 a b - 15360 a b + 10240 a b + 6400 a b - 640 a b 5 7 3 + 1536 a b - 1024 a b - 640 a b + 64 a b) 5 7 3 (-192 c + 128 c + 80 c - 8 c) Proof: F(n) is a C-finite sequence of order, 4 and G(n) is a C-finite sequence of order, 2 Hence their product is a C-finite sequence of order 8 Since the claimed sequence for their product is a C-finite sequence of order, 8, it suffices to check the equality for the first, 16, values The first, 16, members of the sequence F(n) are 2 2 2 2 3 3 3 3 [1, 4 a b, 1 - 4 b - 4 a + 16 a b , 16 a b - 32 a b - 32 a b + 64 a b , 2 2 2 4 2 4 2 4 4 4 2 144 a b - 192 a b - 12 a - 192 a b + 256 a b + 16 a - 12 b 4 3 3 3 5 3 5 3 5 5 + 16 b + 1, 1024 a b - 1024 a b - 192 a b - 1024 a b + 1024 a b 5 3 5 4 4 4 6 + 192 a b - 192 a b + 192 a b + 36 a b, 6400 a b - 5120 a b 4 2 4 6 4 6 6 6 2 6 - 1920 a b + 80 a - 5120 a b + 4096 a b + 1536 a b - 64 a 2 4 2 6 2 2 2 4 6 2 - 1920 a b + 1536 a b + 576 a b - 24 a + 80 b - 64 b - 24 b + 1, 5 5 5 7 5 3 5 7 5 36864 a b - 24576 a b - 15360 a b + 1536 a b - 24576 a b 7 7 7 3 7 3 5 3 7 + 16384 a b + 10240 a b - 1024 a b - 15360 a b + 10240 a b 3 3 3 5 7 3 + 6400 a b - 640 a b + 1536 a b - 1024 a b - 640 a b + 64 a b, 1 2 4 6 8 2 4 6 8 - 40 a + 240 a - 448 a + 256 a - 40 b + 240 b - 448 b + 256 b 2 2 2 4 4 2 4 4 4 6 + 1600 a b - 9600 a b - 9600 a b + 57600 a b - 107520 a b 6 4 6 6 6 2 2 6 6 8 - 107520 a b + 200704 a b + 17920 a b + 17920 a b - 114688 a b 8 6 8 8 8 4 8 2 4 8 - 114688 a b + 65536 a b + 61440 a b - 10240 a b + 61440 a b 2 8 3 3 3 3 - 10240 a b , 100 a b - 1600 a b - 1600 a b + 25600 a b 3 5 5 3 5 5 5 5 - 107520 a b - 107520 a b + 451584 a b + 6720 a b + 6720 a b 5 7 7 5 7 7 7 3 7 - 688128 a b - 688128 a b + 1048576 a b + 163840 a b - 10240 a b 3 7 7 7 9 9 7 9 9 + 163840 a b - 10240 a b - 524288 a b - 524288 a b + 262144 a b 9 5 9 3 9 5 9 3 9 + 344064 a b - 81920 a b + 5120 a b + 344064 a b - 81920 a b 9 2 4 6 8 10 2 + 5120 a b , 1 - 60 a + 560 a - 1792 a + 2304 a - 1024 a - 60 b 4 6 8 10 2 2 2 4 + 560 b - 1792 b + 2304 b - 1024 b + 3600 a b - 33600 a b 4 2 4 4 4 6 6 4 - 33600 a b + 313600 a b - 1003520 a b - 1003520 a b 6 6 6 2 2 6 6 8 + 3211264 a b + 107520 a b + 107520 a b - 4128768 a b 8 6 8 8 8 4 8 2 - 4128768 a b + 5308416 a b + 1290240 a b - 138240 a b 4 8 2 8 8 10 10 8 + 1290240 a b - 138240 a b - 2359296 a b - 2359296 a b 10 10 10 6 10 4 10 2 + 1048576 a b + 1835008 a b - 573440 a b + 61440 a b 6 10 4 10 2 10 3 + 1835008 a b - 573440 a b + 61440 a b , 144 a b - 3360 a b 3 3 3 3 5 5 3 5 5 - 3360 a b + 78400 a b - 501760 a b - 501760 a b + 3211264 a b 5 5 5 7 7 5 7 7 + 21504 a b + 21504 a b - 8257536 a b - 8257536 a b + 21233664 a b 7 3 7 3 7 7 7 9 + 1290240 a b - 55296 a b + 1290240 a b - 55296 a b - 23592960 a b 9 7 9 9 9 5 9 3 - 23592960 a b + 26214400 a b + 9175040 a b - 1433600 a b 9 5 9 3 9 9 + 61440 a b + 9175040 a b - 1433600 a b + 61440 a b 9 11 11 9 11 11 11 7 - 10485760 a b - 10485760 a b + 4194304 a b + 9437184 a b 11 5 11 3 11 7 11 - 3670016 a b + 573440 a b - 24576 a b + 9437184 a b 5 11 3 11 11 2 4 - 3670016 a b + 573440 a b - 24576 a b , 1 - 84 a + 1120 a 6 8 10 2 4 6 8 - 5376 a + 11520 a - 11264 a - 84 b + 1120 b - 5376 b + 11520 b 10 2 2 2 4 4 2 4 4 - 11264 b + 7056 a b - 94080 a b - 94080 a b + 1254400 a b 4 6 6 4 6 6 6 2 - 6021120 a b - 6021120 a b + 28901376 a b + 451584 a b 2 6 6 8 8 6 8 8 + 451584 a b - 61931520 a b - 61931520 a b + 132710400 a b 8 4 8 2 4 8 2 8 + 12902400 a b - 967680 a b + 12902400 a b - 967680 a b 8 10 10 8 10 10 - 129761280 a b - 129761280 a b + 126877696 a b 10 6 10 4 10 2 6 10 + 60555264 a b - 12615680 a b + 946176 a b + 60555264 a b 4 10 2 10 12 6 12 8 - 12615680 a b + 946176 a b - 22020096 a b + 47185920 a b 12 4 12 2 12 6 12 8 + 4587520 a b - 344064 a b - 22020096 b a + 47185920 b a 12 4 12 2 12 12 10 12 + 4587520 b a - 344064 b a + 4096 a + 4096 b - 46137344 a b 12 10 12 12 9 13 13 9 - 46137344 a b + 16777216 a b , 230686720 a b + 230686720 a b 13 7 13 5 13 3 13 - 125829120 a b + 33030144 a b - 3670016 a b + 114688 a b 7 13 5 13 3 13 13 - 125829120 a b + 33030144 a b - 3670016 a b + 114688 a b 3 3 3 3 3 5 + 196 a b - 6272 a b - 6272 a b + 200704 a b - 1806336 a b 5 3 5 5 5 5 - 1806336 a b + 16257024 a b + 56448 a b + 56448 a b 5 7 7 5 7 7 7 3 - 61931520 a b - 61931520 a b + 235929600 a b + 6881280 a b 7 3 7 7 7 9 - 215040 a b + 6881280 a b - 215040 a b - 432537600 a b 9 7 9 9 9 5 9 3 - 432537600 a b + 792985600 a b + 113541120 a b - 12615680 a b 9 5 9 3 9 9 + 394240 a b + 113541120 a b - 12615680 a b + 394240 a b 9 11 11 9 11 11 - 692060160 a b - 692060160 a b + 603979776 a b 11 7 11 5 11 3 11 + 377487360 a b - 99090432 a b + 11010048 a b - 344064 a b 7 11 5 11 3 11 11 + 377487360 a b - 99090432 a b + 11010048 a b - 344064 a b 11 13 13 11 13 13 2 - 201326592 a b - 201326592 a b + 67108864 a b , 1 - 112 a 4 6 8 10 2 4 6 + 2016 a - 13440 a + 42240 a - 67584 a - 112 b + 2016 b - 13440 b 8 10 2 2 2 4 4 2 + 42240 b - 67584 b + 12544 a b - 225792 a b - 225792 a b 4 4 4 6 6 4 6 6 + 4064256 a b - 27095040 a b - 27095040 a b + 180633600 a b 6 2 2 6 6 8 8 6 + 1505280 a b + 1505280 a b - 567705600 a b - 567705600 a b 8 8 8 4 8 2 4 8 + 1784217600 a b + 85155840 a b - 4730880 a b + 85155840 a b 2 8 8 10 10 8 - 4730880 a b - 2854748160 a b - 2854748160 a b 10 10 10 6 10 4 + 4567597056 a b + 908328960 a b - 136249344 a b 10 2 6 10 4 10 2 10 + 7569408 a b + 908328960 a b - 136249344 a b + 7569408 a b 12 6 12 8 12 4 12 2 - 715653120 a b + 2249195520 a b + 107347968 a b - 5963776 a b 14 6 14 8 14 4 14 2 + 220200960 a b - 692060160 a b - 33030144 a b + 1835008 a b 12 6 12 8 12 4 12 2 - 715653120 b a + 2249195520 b a + 107347968 b a - 5963776 b a 14 6 14 8 14 4 14 2 + 220200960 b a - 692060160 b a - 33030144 b a + 1835008 b a 12 14 12 14 10 12 + 53248 a - 16384 a + 53248 b - 16384 b - 3598712832 a b 12 10 12 12 10 14 - 3598712832 a b + 2835349504 a b + 1107296256 a b 12 14 14 10 14 12 - 872415232 a b + 1107296256 a b - 872415232 a b 14 14 9 13 13 9 + 268435456 a b , 12918456320 a b + 12918456320 a b 13 7 13 5 13 3 13 - 4844421120 a b + 924844032 a b - 77070336 a b + 1835008 a b 7 13 5 13 3 13 13 - 4844421120 a b + 924844032 a b - 77070336 a b + 1835008 a b 3 3 3 3 3 5 + 256 a b - 10752 a b - 10752 a b + 451584 a b - 5419008 a b 5 3 5 5 5 5 - 5419008 a b + 65028096 a b + 129024 a b + 129024 a b 5 7 7 5 7 7 7 3 - 340623360 a b - 340623360 a b + 1784217600 a b + 28385280 a b 7 3 7 7 7 9 - 675840 a b + 28385280 a b - 675840 a b - 4757913600 a b 9 7 9 9 9 5 9 3 - 4757913600 a b + 12687769600 a b + 908328960 a b - 75694080 a b 9 5 9 3 9 9 + 1802240 a b + 908328960 a b - 75694080 a b + 1802240 a b 9 11 11 9 11 11 - 17993564160 a b - 17993564160 a b + 25518145536 a b 11 7 11 5 11 3 11 + 6747586560 a b - 1288175616 a b + 107347968 a b - 2555904 a b 7 11 5 11 3 11 11 + 6747586560 a b - 1288175616 a b + 107347968 a b - 2555904 a b 15 7 15 5 15 3 15 + 1384120320 a b - 264241152 a b + 22020096 a b - 524288 a b 7 15 5 15 3 15 15 + 1384120320 a b - 264241152 a b + 22020096 a b - 524288 a b 11 13 13 11 13 13 - 18320719872 a b - 18320719872 a b + 13153337344 a b 15 9 9 15 11 15 - 3690987520 a b - 3690987520 a b + 5234491392 a b 13 15 15 11 15 13 - 3758096384 a b + 5234491392 a b - 3758096384 a b 15 15 + 1073741824 a b ] The first, 16, members of the sequence G(n) are 2 3 2 4 3 5 [1, 2 c, -1 + 4 c , -4 c + 8 c , -12 c + 16 c + 1, -32 c + 32 c + 6 c, 4 6 2 5 7 3 -80 c + 64 c + 24 c - 1, -192 c + 128 c + 80 c - 8 c, 6 8 4 2 -448 c + 256 c + 240 c - 40 c + 1, 7 9 5 3 -1024 c + 512 c + 672 c - 160 c + 10 c, 8 10 6 4 2 -2304 c + 1024 c + 1792 c - 560 c + 60 c - 1, 9 11 7 5 3 -5120 c + 2048 c + 4608 c - 1792 c + 280 c - 12 c, 10 12 8 6 4 2 -11264 c + 4096 c + 11520 c - 5376 c + 1120 c - 84 c + 1, 11 13 9 7 5 3 -24576 c + 8192 c + 28160 c - 15360 c + 4032 c - 448 c + 14 c, 12 14 10 8 6 4 2 -53248 c + 16384 c + 67584 c - 42240 c + 13440 c - 2016 c + 112 c 13 15 11 9 7 5 - 1, -114688 c + 32768 c + 159744 c - 112640 c + 42240 c - 8064 c 3 + 672 c - 16 c] The first, 16, members of the propsed product are [1, 8 a b c, 2 2 2 2 2 2 2 2 2 2 2 2 4 a - 16 a c + 4 b - 16 b c - 1 + 4 c - 16 a b + 64 a b c , 3 3 3 3 3 -64 a b c + 128 a b c + 128 a b c - 256 a b c + 128 a b c 3 3 3 3 3 3 3 2 4 2 - 256 a b c - 256 a b c + 512 a b c , 1 - 12 a + 16 a - 12 b 4 2 2 2 4 4 2 4 4 2 4 + 16 b + 144 a b - 192 a b - 192 a b + 256 a b - 12 c + 16 c 2 2 2 2 2 4 2 4 2 2 4 4 - 1728 a b c + 2304 a b c + 2304 a b c - 3072 a b c 4 2 2 4 2 4 4 4 2 4 4 4 + 2304 a b c - 3072 a b c - 3072 a b c + 4096 a b c 2 2 2 4 4 2 4 4 2 2 2 4 + 144 a c - 192 a c - 192 a c + 256 a c + 144 b c - 192 b c 4 2 4 4 3 3 - 192 b c + 256 b c , 216 a b c - 1152 a b c - 1152 a b c 3 3 3 3 3 3 3 + 6144 a b c - 1152 a b c + 6144 a b c + 6144 a b c 3 3 3 3 3 5 3 5 3 3 5 5 - 32768 a b c + 32768 a b c + 32768 a b c - 32768 a b c 3 5 3 5 5 3 3 5 3 5 - 6144 a b c - 6144 a b c + 32768 a b c - 32768 a b c 5 3 5 5 3 5 5 5 5 5 - 6144 a b c - 32768 a b c + 32768 a b c + 6144 a b c 5 3 5 5 5 3 5 5 3 - 6144 a b c + 6144 a b c + 1152 a b c - 6144 a b c - 6144 a b c 5 5 5 5 2 4 6 + 6144 a b c + 1152 a b c + 1152 a b c , -1 + 24 a - 80 a + 64 a 2 4 6 2 2 2 4 4 2 4 4 + 24 b - 80 b + 64 b - 576 a b + 1920 a b + 1920 a b - 6400 a b 4 6 6 4 6 6 6 2 2 6 + 5120 a b + 5120 a b - 4096 a b - 1536 a b - 1536 a b 4 6 4 2 4 6 2 2 2 + 409600 a b c + 24 c - 80 c + 64 c + 13824 a b c 2 2 4 2 4 2 2 4 4 4 2 2 - 46080 a b c - 46080 a b c + 153600 a b c - 46080 a b c 4 2 4 4 4 2 4 4 4 4 4 6 + 153600 a b c + 153600 a b c - 512000 a b c + 409600 a b c 2 2 2 4 4 2 4 4 2 2 - 576 a c + 1920 a c + 1920 a c - 6400 a c - 576 b c 2 4 4 2 4 4 4 6 6 4 + 1920 b c + 1920 b c - 6400 b c + 5120 a c + 5120 a c 6 6 6 2 2 6 4 6 6 4 - 4096 a c - 1536 a c - 1536 a c + 5120 b c + 5120 b c 6 6 6 2 2 6 4 6 6 4 6 2 - 4096 b c - 1536 b c - 1536 b c - 327680 a b c - 122880 a b c 4 2 6 6 4 4 6 4 6 6 4 2 - 122880 a b c + 409600 a b c - 327680 a b c - 122880 a b c 6 6 4 6 6 6 6 6 2 6 2 4 - 327680 a b c + 262144 a b c + 98304 a b c - 122880 a b c 6 2 6 6 2 2 2 4 6 2 6 4 + 98304 a b c + 36864 a b c - 122880 a b c - 122880 a b c 2 6 6 2 6 2 2 2 6 + 98304 a b c + 36864 a b c + 36864 a b c , -512 a b c 3 3 3 3 3 3 3 + 5120 a b c + 5120 a b c - 51200 a b c + 5120 a b c - 51200 a b c 3 3 3 3 3 3 3 5 3 5 3 - 51200 a b c + 512000 a b c - 1228800 a b c - 1228800 a b c 3 5 5 3 5 3 5 5 3 3 + 2949120 a b c + 122880 a b c + 122880 a b c - 1228800 a b c 5 3 5 5 3 5 5 3 5 5 5 + 2949120 a b c + 122880 a b c + 2949120 a b c - 7077888 a b c 5 5 5 3 5 5 5 - 294912 a b c + 122880 a b c - 294912 a b c - 12288 a b c 3 5 5 3 5 5 5 + 122880 a b c + 122880 a b c - 294912 a b c - 12288 a b c 5 5 5 7 5 7 5 5 7 7 - 12288 a b c + 4718592 a b c + 4718592 a b c - 3145728 a b c 5 7 3 5 7 5 3 7 5 7 - 1966080 a b c + 196608 a b c - 1966080 a b c + 196608 a b c 7 5 5 7 5 7 7 5 3 7 5 + 4718592 a b c - 3145728 a b c - 1966080 a b c + 196608 a b c 7 7 5 7 7 7 7 7 3 7 7 - 3145728 a b c + 2097152 a b c + 1310720 a b c - 131072 a b c 7 3 5 7 3 7 7 3 3 7 3 - 1966080 a b c + 1310720 a b c + 819200 a b c - 81920 a b c 7 5 7 7 7 3 7 + 196608 a b c - 131072 a b c - 81920 a b c + 8192 a b c 3 5 7 3 7 5 3 7 7 3 7 3 - 1966080 a b c - 1966080 a b c + 1310720 a b c + 819200 a b c 3 7 3 3 7 3 7 5 7 - 81920 a b c + 819200 a b c - 81920 a b c + 196608 a b c 7 5 7 7 7 3 7 + 196608 a b c - 131072 a b c - 81920 a b c + 8192 a b c 3 7 7 2 4 6 8 - 81920 a b c + 8192 a b c , 1 - 40 a + 240 a - 448 a + 256 a 2 4 6 8 2 2 2 4 4 2 - 40 b + 240 b - 448 b + 256 b + 1600 a b - 9600 a b - 9600 a b 4 4 4 6 6 4 6 6 6 2 + 57600 a b - 107520 a b - 107520 a b + 200704 a b + 17920 a b 2 6 6 8 8 6 8 8 8 4 + 17920 a b - 114688 a b - 114688 a b + 65536 a b + 61440 a b 8 2 4 8 2 8 4 6 4 2 - 10240 a b + 61440 a b - 10240 a b - 25804800 a b c - 40 c 4 6 8 2 2 2 2 2 4 + 240 c - 448 c + 256 c - 64000 a b c + 384000 a b c 2 4 2 2 4 4 4 2 2 4 2 4 + 384000 a b c - 2304000 a b c + 384000 a b c - 2304000 a b c 4 4 2 4 4 4 4 4 6 2 2 - 2304000 a b c + 13824000 a b c - 25804800 a b c + 1600 a c 2 4 4 2 4 4 2 2 2 4 - 9600 a c - 9600 a c + 57600 a c + 1600 b c - 9600 b c 4 2 4 4 4 6 6 4 6 6 - 9600 b c + 57600 b c - 107520 a c - 107520 a c + 200704 a c 6 2 2 6 4 6 6 4 6 6 + 17920 a c + 17920 a c - 107520 b c - 107520 b c + 200704 b c 6 2 2 6 2 8 4 8 6 8 + 17920 b c + 17920 b c - 10240 a c + 61440 a c - 114688 a c 8 6 8 8 8 4 8 2 2 8 - 114688 a c + 65536 a c + 61440 a c - 10240 a c - 10240 b c 4 8 6 8 8 6 8 8 8 4 + 61440 b c - 114688 b c - 114688 b c + 65536 b c + 61440 b c 8 2 4 6 6 4 6 2 4 2 6 - 10240 b c + 48168960 a b c + 4300800 a b c + 4300800 a b c 6 4 4 6 4 6 6 4 2 - 25804800 a b c + 48168960 a b c + 4300800 a b c 6 6 4 6 6 6 6 6 2 + 48168960 a b c - 89915392 a b c - 8028160 a b c 6 2 4 6 2 6 6 2 2 2 4 6 + 4300800 a b c - 8028160 a b c - 716800 a b c + 4300800 a b c 2 6 4 2 6 6 2 6 2 2 2 6 + 4300800 a b c - 8028160 a b c - 716800 a b c - 716800 a b c 2 2 8 2 4 8 4 2 8 + 409600 a b c - 2457600 a b c - 2457600 a b c 4 4 8 4 6 8 6 4 8 + 14745600 a b c - 27525120 a b c - 27525120 a b c 6 6 8 6 2 8 2 6 8 + 51380224 a b c + 4587520 a b c + 4587520 a b c 6 8 6 6 8 8 6 8 4 + 51380224 a b c - 29360128 a b c - 27525120 a b c 6 8 2 8 6 6 8 6 8 + 4587520 a b c + 51380224 a b c - 29360128 a b c 8 6 4 8 6 2 8 8 6 - 27525120 a b c + 4587520 a b c - 29360128 a b c 8 8 8 8 8 4 8 8 2 + 16777216 a b c + 15728640 a b c - 2621440 a b c 8 4 6 8 4 8 8 4 4 - 27525120 a b c + 15728640 a b c + 14745600 a b c 8 4 2 8 2 6 8 2 8 - 2457600 a b c + 4587520 a b c - 2621440 a b c 8 2 4 8 2 2 4 8 6 - 2457600 a b c + 409600 a b c - 27525120 a b c 4 8 8 4 8 4 4 8 2 + 15728640 a b c + 14745600 a b c - 2457600 a b c 2 8 6 2 8 8 2 8 4 2 8 2 + 4587520 a b c - 2621440 a b c - 2457600 a b c + 409600 a b c 3 3 3 3 3 , 1000 a b c - 16000 a b c - 16000 a b c + 256000 a b c - 16000 a b c 3 3 3 3 3 3 3 3 3 5 + 256000 a b c + 256000 a b c - 4096000 a b c + 17203200 a b c 3 5 3 3 5 5 3 5 + 17203200 a b c - 72253440 a b c - 1075200 a b c 3 5 5 3 3 5 3 5 - 1075200 a b c + 17203200 a b c - 72253440 a b c 5 3 5 5 3 5 5 5 - 1075200 a b c - 72253440 a b c + 303464448 a b c 5 5 5 3 5 5 5 + 4515840 a b c - 1075200 a b c + 4515840 a b c + 67200 a b c 3 5 5 3 5 5 5 - 1075200 a b c - 1075200 a b c + 4515840 a b c + 67200 a b c 5 9 3 9 3 9 + 67200 a b c + 51200 a b c - 819200 a b c - 819200 a b c 3 3 9 3 5 9 5 3 9 + 13107200 a b c - 55050240 a b c - 55050240 a b c 5 5 9 5 9 5 9 + 231211008 a b c + 3440640 a b c + 3440640 a b c 5 7 9 7 5 9 7 7 9 - 352321536 a b c - 352321536 a b c + 536870912 a b c 7 3 9 7 9 3 7 9 + 83886080 a b c - 5242880 a b c + 83886080 a b c 7 9 7 9 7 7 9 9 - 5242880 a b c + 536870912 a b c - 268435456 a b c 7 9 5 7 9 3 7 9 - 352321536 a b c + 83886080 a b c - 5242880 a b c 9 7 7 9 7 9 9 7 5 + 536870912 a b c - 268435456 a b c - 352321536 a b c 9 7 3 9 7 9 9 7 + 83886080 a b c - 5242880 a b c - 268435456 a b c 9 9 9 9 9 5 9 9 3 + 134217728 a b c + 176160768 a b c - 41943040 a b c 9 9 9 5 7 9 5 9 + 2621440 a b c - 352321536 a b c + 176160768 a b c 9 5 5 9 5 3 9 5 + 231211008 a b c - 55050240 a b c + 3440640 a b c 9 3 7 9 3 9 9 3 5 + 83886080 a b c - 41943040 a b c - 55050240 a b c 9 3 3 9 3 9 7 9 9 + 13107200 a b c - 819200 a b c - 5242880 a b c + 2621440 a b c 9 5 9 3 9 5 9 7 + 3440640 a b c - 819200 a b c + 51200 a b c - 352321536 a b c 5 9 9 5 9 5 5 9 3 + 176160768 a b c + 231211008 a b c - 55050240 a b c 5 9 3 9 7 3 9 9 + 3440640 a b c + 83886080 a b c - 41943040 a b c 3 9 5 3 9 3 3 9 9 7 - 55050240 a b c + 13107200 a b c - 819200 a b c - 5242880 a b c 9 9 9 5 9 3 9 + 2621440 a b c + 3440640 a b c - 819200 a b c + 51200 a b c 5 5 7 5 7 5 5 7 7 - 462422016 a b c - 462422016 a b c + 704643072 a b c 5 7 3 5 7 5 3 7 + 110100480 a b c - 6881280 a b c + 110100480 a b c 5 7 7 5 5 7 5 7 - 6881280 a b c - 462422016 a b c + 704643072 a b c 7 5 3 7 5 7 7 5 + 110100480 a b c - 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3255450992640 a b c + 77510737920 a b c - 3255450992640 a b c 7 13 7 9 13 + 77510737920 a b c + 545675594956800 a b c 9 7 13 9 9 13 + 545675594956800 a b c - 1455134919884800 a b c 9 5 13 9 3 13 - 104174431764480 a b c + 8681202647040 a b c 9 13 5 9 13 - 206695301120 a b c - 104174431764480 a b c 3 9 13 9 13 + 8681202647040 a b c - 206695301120 a b c 9 11 13 11 9 13 + 2063645886382080 a b c + 2063645886382080 a b c 11 11 13 11 7 13 - 2926625075232768 a b c - 773867207393280 a b c 11 5 13 11 3 13 + 147738285047808 a b c - 12311523753984 a b c 11 13 7 11 13 + 293131517952 a b c - 773867207393280 a b c 5 11 13 3 11 13 + 147738285047808 a b c - 12311523753984 a b c 11 13 11 13 11 + 293131517952 a b c - 2926625075232768 a b c 11 13 13 11 13 9 + 2101166720679936 a b c + 2063645886382080 a b c 11 13 7 11 13 5 - 773867207393280 a b c + 147738285047808 a b c 11 13 3 11 13 - 12311523753984 a b c + 293131517952 a b c 13 11 11 13 11 13 - 2926625075232768 a b c + 2101166720679936 a b c 13 11 9 13 11 7 + 2063645886382080 a b c - 773867207393280 a b c 13 11 5 13 11 3 + 147738285047808 a b c - 12311523753984 a b c 13 11 13 13 11 + 293131517952 a b c + 2101166720679936 a b c 13 13 13 13 13 9 - 1508529953308672 a b c - 1481591918428160 a b c 13 13 7 13 13 5 + 555596969410560 a b c - 106068512342016 a b c 13 13 3 13 13 + 8839042695168 a b c - 210453397504 a b c 9 13 15 13 9 15 + 423311976693760 a b c + 423311976693760 a b c 13 7 15 13 5 15 - 158741991260160 a b c + 30305289240576 a b c 13 3 15 13 15 - 2525440770048 a b c + 60129542144 a b c 7 13 15 5 13 15 - 158741991260160 a b c + 30305289240576 a b c 3 13 15 13 15 15 - 2525440770048 a b c + 60129542144 a b c + 8388608 a b c 3 15 3 15 3 3 15 - 352321536 a b c - 352321536 a b c + 14797504512 a b c 3 5 15 5 3 15 - 177570054144 a b c - 177570054144 a b c 5 5 15 5 15 5 15 + 2130840649728 a b c + 4227858432 a b c + 4227858432 a b c 5 7 15 7 5 15 - 11161546260480 a b c - 11161546260480 a b c 7 7 15 7 3 15 7 15 + 58465242316800 a b c + 930128855040 a b c - 22145925120 a b c 3 7 15 7 15 + 930128855040 a b c - 22145925120 a b c 7 9 15 9 7 15 - 155907312844800 a b c - 155907312844800 a b c 9 9 15 9 5 15 + 415752834252800 a b c + 29764123361280 a b c 9 3 15 9 15 - 2480343613440 a b c + 59055800320 a b c 5 9 15 3 9 15 + 29764123361280 a b c - 2480343613440 a b c 9 15 9 11 15 + 59055800320 a b c - 589613110394880 a b c 11 9 15 11 11 15 - 589613110394880 a b c + 836178592923648 a b c 11 7 15 11 5 15 + 221104916398080 a b c - 42210938585088 a b c 11 3 15 11 15 + 3517578215424 a b c - 83751862272 a b c 7 11 15 5 11 15 + 221104916398080 a b c - 42210938585088 a b c 3 11 15 11 15 + 3517578215424 a b c - 83751862272 a b c 15 7 13 15 7 15 - 158741991260160 a b c + 45354854645760 a b c 15 7 11 15 7 9 + 221104916398080 a b c - 155907312844800 a b c 15 7 7 15 7 5 + 58465242316800 a b c - 11161546260480 a b c 15 7 3 15 7 + 930128855040 a b c - 22145925120 a b c 15 5 13 15 5 15 + 30305289240576 a b c - 8658654068736 a b c 15 5 11 15 5 9 - 42210938585088 a b c + 29764123361280 a b c 15 5 7 15 5 5 - 11161546260480 a b c + 2130840649728 a b c 15 5 3 15 5 15 3 13 - 177570054144 a b c + 4227858432 a b c - 2525440770048 a b c 15 3 15 15 3 11 + 721554505728 a b c + 3517578215424 a b c 15 3 9 15 3 7 - 2480343613440 a b c + 930128855040 a b c 15 3 5 15 3 3 15 3 - 177570054144 a b c + 14797504512 a b c - 352321536 a b c 15 13 15 15 15 11 + 60129542144 a b c - 17179869184 a b c - 83751862272 a b c 15 9 15 7 15 5 + 59055800320 a b c - 22145925120 a b c + 4227858432 a b c 15 3 15 7 15 13 - 352321536 a b c + 8388608 a b c - 158741991260160 a b c 7 15 15 7 15 11 + 45354854645760 a b c + 221104916398080 a b c 7 15 9 7 15 7 - 155907312844800 a b c + 58465242316800 a b c 7 15 5 7 15 3 7 15 - 11161546260480 a b c + 930128855040 a b c - 22145925120 a b c 5 15 13 5 15 15 + 30305289240576 a b c - 8658654068736 a b c 5 15 11 5 15 9 - 42210938585088 a b c + 29764123361280 a b c 5 15 7 5 15 5 - 11161546260480 a b c + 2130840649728 a b c 5 15 3 5 15 3 15 13 - 177570054144 a b c + 4227858432 a b c - 2525440770048 a b c 3 15 15 3 15 11 + 721554505728 a b c + 3517578215424 a b c 3 15 9 3 15 7 - 2480343613440 a b c + 930128855040 a b c 3 15 5 3 15 3 3 15 - 177570054144 a b c + 14797504512 a b c - 352321536 a b c 15 13 15 15 15 11 + 60129542144 a b c - 17179869184 a b c - 83751862272 a b c 15 9 15 7 15 5 + 59055800320 a b c - 22145925120 a b c + 4227858432 a b c 15 3 15 11 13 15 - 352321536 a b c + 8388608 a b c - 600333348765696 a b c 13 11 15 13 13 15 - 600333348765696 a b c + 431008558088192 a b c 15 9 13 15 9 15 + 423311976693760 a b c - 120946279055360 a b c 15 9 11 15 9 9 - 589613110394880 a b c + 415752834252800 a b c 15 9 7 15 9 5 - 155907312844800 a b c + 29764123361280 a b c 15 9 3 15 9 - 2480343613440 a b c + 59055800320 a b c 9 15 13 9 15 15 + 423311976693760 a b c - 120946279055360 a b c 9 15 11 9 15 9 - 589613110394880 a b c + 415752834252800 a b c 9 15 7 9 15 5 - 155907312844800 a b c + 29764123361280 a b c 9 15 3 9 15 - 2480343613440 a b c + 59055800320 a b c 11 15 13 11 15 15 - 600333348765696 a b c + 171523813933056 a b c 11 15 11 11 15 9 + 836178592923648 a b c - 589613110394880 a b c 11 15 7 11 15 5 + 221104916398080 a b c - 42210938585088 a b c 11 15 3 11 15 + 3517578215424 a b c - 83751862272 a b c 13 15 13 13 15 15 + 431008558088192 a b c - 123145302310912 a b c 13 15 11 13 15 9 - 600333348765696 a b c + 423311976693760 a b c 13 15 7 13 15 5 - 158741991260160 a b c + 30305289240576 a b c 13 15 3 13 15 - 2525440770048 a b c + 60129542144 a b c 15 11 13 15 11 15 - 600333348765696 a b c + 171523813933056 a b c 15 11 11 15 11 9 + 836178592923648 a b c - 589613110394880 a b c 15 11 7 15 11 5 + 221104916398080 a b c - 42210938585088 a b c 15 11 3 15 11 + 3517578215424 a b c - 83751862272 a b c 15 13 13 15 13 15 + 431008558088192 a b c - 123145302310912 a b c 15 13 11 15 13 9 - 600333348765696 a b c + 423311976693760 a b c 15 13 7 15 13 5 - 158741991260160 a b c + 30305289240576 a b c 15 13 3 15 13 - 2525440770048 a b c + 60129542144 a b c 15 15 13 15 15 15 - 123145302310912 a b c + 35184372088832 a b c 15 15 11 15 15 9 + 171523813933056 a b c - 120946279055360 a b c 15 15 7 15 15 5 + 45354854645760 a b c - 8658654068736 a b c 15 15 3 15 15 + 721554505728 a b c - 17179869184 a b c] 1, times , 1, is indeed , 1 4 a b, times , 2 c, is indeed , 8 a b c 2 2 2 2 2 1 - 4 b - 4 a + 16 a b , times , -1 + 4 c , is indeed , 2 2 2 2 2 2 2 2 2 2 2 2 4 a - 16 a c + 4 b - 16 b c - 1 + 4 c - 16 a b + 64 a b c 3 3 3 3 3 16 a b - 32 a b - 32 a b + 64 a b , times , -4 c + 8 c , is indeed , 3 3 3 3 3 -64 a b c + 128 a b c + 128 a b c - 256 a b c + 128 a b c 3 3 3 3 3 3 3 - 256 a b c - 256 a b c + 512 a b c 2 2 2 4 2 4 2 4 4 4 2 4 144 a b - 192 a b - 12 a - 192 a b + 256 a b + 16 a - 12 b + 16 b 2 4 2 4 2 + 1, times , -12 c + 16 c + 1, is indeed , 1 - 12 a + 16 a - 12 b 4 2 2 2 4 4 2 4 4 2 4 + 16 b + 144 a b - 192 a b - 192 a b + 256 a b - 12 c + 16 c 2 2 2 2 2 4 2 4 2 2 4 4 - 1728 a b c + 2304 a b c + 2304 a b c - 3072 a b c 4 2 2 4 2 4 4 4 2 4 4 4 + 2304 a b c - 3072 a b c - 3072 a b c + 4096 a b c 2 2 2 4 4 2 4 4 2 2 2 4 + 144 a c - 192 a c - 192 a c + 256 a c + 144 b c - 192 b c 4 2 4 4 - 192 b c + 256 b c 3 3 3 5 3 5 3 5 5 5 1024 a b - 1024 a b - 192 a b - 1024 a b + 1024 a b + 192 a b 3 5 3 5 - 192 a b + 192 a b + 36 a b, times , -32 c + 32 c + 6 c, is indeed , 3 3 3 3 3 216 a b c - 1152 a b c - 1152 a b c + 6144 a b c - 1152 a b c 3 3 3 3 3 3 3 3 3 5 + 6144 a b c + 6144 a b c - 32768 a b c + 32768 a b c 3 5 3 3 5 5 3 5 3 5 + 32768 a b c - 32768 a b c - 6144 a b c - 6144 a b c 5 3 3 5 3 5 5 3 5 5 3 + 32768 a b c - 32768 a b c - 6144 a b c - 32768 a b c 5 5 5 5 5 5 3 5 5 + 32768 a b c + 6144 a b c - 6144 a b c + 6144 a b c 5 3 5 5 3 5 5 5 + 1152 a b c - 6144 a b c - 6144 a b c + 6144 a b c + 1152 a b c 5 + 1152 a b c 4 4 4 6 4 2 4 6 4 6 6 6400 a b - 5120 a b - 1920 a b + 80 a - 5120 a b + 4096 a b 6 2 6 2 4 2 6 2 2 2 4 + 1536 a b - 64 a - 1920 a b + 1536 a b + 576 a b - 24 a + 80 b 6 2 4 6 2 - 64 b - 24 b + 1, times , -80 c + 64 c + 24 c - 1, is indeed , -1 2 4 6 2 4 6 2 2 2 4 + 24 a - 80 a + 64 a + 24 b - 80 b + 64 b - 576 a b + 1920 a b 4 2 4 4 4 6 6 4 6 6 + 1920 a b - 6400 a b + 5120 a b + 5120 a b - 4096 a b 6 2 2 6 4 6 4 2 4 6 - 1536 a b - 1536 a b + 409600 a b c + 24 c - 80 c + 64 c 2 2 2 2 2 4 2 4 2 2 4 4 + 13824 a b c - 46080 a b c - 46080 a b c + 153600 a b c 4 2 2 4 2 4 4 4 2 4 4 4 - 46080 a b c + 153600 a b c + 153600 a b c - 512000 a b c 4 4 6 2 2 2 4 4 2 4 4 + 409600 a b c - 576 a c + 1920 a c + 1920 a c - 6400 a c 2 2 2 4 4 2 4 4 4 6 - 576 b c + 1920 b c + 1920 b c - 6400 b c + 5120 a c 6 4 6 6 6 2 2 6 4 6 + 5120 a c - 4096 a c - 1536 a c - 1536 a c + 5120 b c 6 4 6 6 6 2 2 6 4 6 6 + 5120 b c - 4096 b c - 1536 b c - 1536 b c - 327680 a b c 4 6 2 4 2 6 6 4 4 6 4 6 - 122880 a b c - 122880 a b c + 409600 a b c - 327680 a b c 6 4 2 6 6 4 6 6 6 6 6 2 - 122880 a b c - 327680 a b c + 262144 a b c + 98304 a b c 6 2 4 6 2 6 6 2 2 2 4 6 - 122880 a b c + 98304 a b c + 36864 a b c - 122880 a b c 2 6 4 2 6 6 2 6 2 2 2 6 - 122880 a b c + 98304 a b c + 36864 a b c + 36864 a b c 5 5 5 7 5 3 5 7 5 7 7 36864 a b - 24576 a b - 15360 a b + 1536 a b - 24576 a b + 16384 a b 7 3 7 3 5 3 7 3 3 + 10240 a b - 1024 a b - 15360 a b + 10240 a b + 6400 a b 3 5 7 3 - 640 a b + 1536 a b - 1024 a b - 640 a b + 64 a b, times , 5 7 3 3 -192 c + 128 c + 80 c - 8 c, is indeed , -512 a b c + 5120 a b c 3 3 3 3 3 3 + 5120 a b c - 51200 a b c + 5120 a b c - 51200 a b c 3 3 3 3 3 3 3 5 3 5 3 - 51200 a b c + 512000 a b c - 1228800 a b c - 1228800 a b c 3 5 5 3 5 3 5 5 3 3 + 2949120 a b c + 122880 a b c + 122880 a b c - 1228800 a b c 5 3 5 5 3 5 5 3 5 5 5 + 2949120 a b c + 122880 a b c + 2949120 a b c - 7077888 a b c 5 5 5 3 5 5 5 - 294912 a b c + 122880 a b c - 294912 a b c - 12288 a b c 3 5 5 3 5 5 5 + 122880 a b c + 122880 a b c - 294912 a b c - 12288 a b c 5 5 5 7 5 7 5 5 7 7 - 12288 a b c + 4718592 a b c + 4718592 a b c - 3145728 a b c 5 7 3 5 7 5 3 7 5 7 - 1966080 a b c + 196608 a b c - 1966080 a b c + 196608 a b c 7 5 5 7 5 7 7 5 3 7 5 + 4718592 a b c - 3145728 a b c - 1966080 a b c + 196608 a b c 7 7 5 7 7 7 7 7 3 7 7 - 3145728 a b c + 2097152 a b c + 1310720 a b c - 131072 a b c 7 3 5 7 3 7 7 3 3 7 3 - 1966080 a b c + 1310720 a b c + 819200 a b c - 81920 a b c 7 5 7 7 7 3 7 + 196608 a b c - 131072 a b c - 81920 a b c + 8192 a b c 3 5 7 3 7 5 3 7 7 3 7 3 - 1966080 a b c - 1966080 a b c + 1310720 a b c + 819200 a b c 3 7 3 3 7 3 7 5 7 - 81920 a b c + 819200 a b c - 81920 a b c + 196608 a b c 7 5 7 7 7 3 7 + 196608 a b c - 131072 a b c - 81920 a b c + 8192 a b c 3 7 7 - 81920 a b c + 8192 a b c 2 4 6 8 2 4 6 8 1 - 40 a + 240 a - 448 a + 256 a - 40 b + 240 b - 448 b + 256 b 2 2 2 4 4 2 4 4 4 6 + 1600 a b - 9600 a b - 9600 a b + 57600 a b - 107520 a b 6 4 6 6 6 2 2 6 6 8 - 107520 a b + 200704 a b + 17920 a b + 17920 a b - 114688 a b 8 6 8 8 8 4 8 2 4 8 - 114688 a b + 65536 a b + 61440 a b - 10240 a b + 61440 a b 2 8 6 8 4 2 - 10240 a b , times , -448 c + 256 c + 240 c - 40 c + 1, is indeed , 1 2 4 6 8 2 4 6 8 - 40 a + 240 a - 448 a + 256 a - 40 b + 240 b - 448 b + 256 b 2 2 2 4 4 2 4 4 4 6 + 1600 a b - 9600 a b - 9600 a b + 57600 a b - 107520 a b 6 4 6 6 6 2 2 6 6 8 - 107520 a b + 200704 a b + 17920 a b + 17920 a b - 114688 a b 8 6 8 8 8 4 8 2 4 8 - 114688 a b + 65536 a b + 61440 a b - 10240 a b + 61440 a b 2 8 4 6 4 2 4 6 8 - 10240 a b - 25804800 a b c - 40 c + 240 c - 448 c + 256 c 2 2 2 2 2 4 2 4 2 2 4 4 - 64000 a b c + 384000 a b c + 384000 a b c - 2304000 a b c 4 2 2 4 2 4 4 4 2 + 384000 a b c - 2304000 a b c - 2304000 a b c 4 4 4 4 4 6 2 2 2 4 + 13824000 a b c - 25804800 a b c + 1600 a c - 9600 a c 4 2 4 4 2 2 2 4 4 2 - 9600 a c + 57600 a c + 1600 b c - 9600 b c - 9600 b c 4 4 4 6 6 4 6 6 6 2 + 57600 b c - 107520 a c - 107520 a c + 200704 a c + 17920 a c 2 6 4 6 6 4 6 6 6 2 + 17920 a c - 107520 b c - 107520 b c + 200704 b c + 17920 b c 2 6 2 8 4 8 6 8 8 6 + 17920 b c - 10240 a c + 61440 a c - 114688 a c - 114688 a c 8 8 8 4 8 2 2 8 4 8 + 65536 a c + 61440 a c - 10240 a c - 10240 b c + 61440 b c 6 8 8 6 8 8 8 4 8 2 - 114688 b c - 114688 b c + 65536 b c + 61440 b c - 10240 b c 4 6 6 4 6 2 4 2 6 + 48168960 a b c + 4300800 a b c + 4300800 a b c 6 4 4 6 4 6 6 4 2 - 25804800 a b c + 48168960 a b c + 4300800 a b c 6 6 4 6 6 6 6 6 2 + 48168960 a b c - 89915392 a b c - 8028160 a b c 6 2 4 6 2 6 6 2 2 2 4 6 + 4300800 a b c - 8028160 a b c - 716800 a b c + 4300800 a b c 2 6 4 2 6 6 2 6 2 2 2 6 + 4300800 a b c - 8028160 a b c - 716800 a b c - 716800 a b c 2 2 8 2 4 8 4 2 8 + 409600 a b c - 2457600 a b c - 2457600 a b c 4 4 8 4 6 8 6 4 8 + 14745600 a b c - 27525120 a b c - 27525120 a b c 6 6 8 6 2 8 2 6 8 + 51380224 a b c + 4587520 a b c + 4587520 a b c 6 8 6 6 8 8 6 8 4 + 51380224 a b c - 29360128 a b c - 27525120 a b c 6 8 2 8 6 6 8 6 8 + 4587520 a b c + 51380224 a b c - 29360128 a b c 8 6 4 8 6 2 8 8 6 - 27525120 a b c + 4587520 a b c - 29360128 a b c 8 8 8 8 8 4 8 8 2 + 16777216 a b c + 15728640 a b c - 2621440 a b c 8 4 6 8 4 8 8 4 4 - 27525120 a b c + 15728640 a b c + 14745600 a b c 8 4 2 8 2 6 8 2 8 - 2457600 a b c + 4587520 a b c - 2621440 a b c 8 2 4 8 2 2 4 8 6 - 2457600 a b c + 409600 a b c - 27525120 a b c 4 8 8 4 8 4 4 8 2 + 15728640 a b c + 14745600 a b c - 2457600 a b c 2 8 6 2 8 8 2 8 4 2 8 2 + 4587520 a b c - 2621440 a b c - 2457600 a b c + 409600 a b c 3 3 3 3 3 5 5 3 100 a b - 1600 a b - 1600 a b + 25600 a b - 107520 a b - 107520 a b 5 5 5 5 5 7 7 5 + 451584 a b + 6720 a b + 6720 a b - 688128 a b - 688128 a b 7 7 7 3 7 3 7 7 + 1048576 a b + 163840 a b - 10240 a b + 163840 a b - 10240 a b 7 9 9 7 9 9 9 5 9 3 - 524288 a b - 524288 a b + 262144 a b + 344064 a b - 81920 a b 9 5 9 3 9 9 + 5120 a b + 344064 a b - 81920 a b + 5120 a b , times , 7 9 5 3 -1024 c + 512 c + 672 c - 160 c + 10 c, is indeed , 1000 a b c 3 3 3 3 3 - 16000 a b c - 16000 a b c + 256000 a b c - 16000 a b c 3 3 3 3 3 3 3 3 3 5 + 256000 a b c + 256000 a b c - 4096000 a b c + 17203200 a b c 3 5 3 3 5 5 3 5 + 17203200 a b c - 72253440 a b c - 1075200 a b c 3 5 5 3 3 5 3 5 - 1075200 a b c + 17203200 a b c - 72253440 a b c 5 3 5 5 3 5 5 5 - 1075200 a b c - 72253440 a b c + 303464448 a b c 5 5 5 3 5 5 5 + 4515840 a b c - 1075200 a b c + 4515840 a b c + 67200 a b c 3 5 5 3 5 5 5 - 1075200 a b c - 1075200 a b c + 4515840 a b c + 67200 a b c 5 9 3 9 3 9 + 67200 a b c + 51200 a b c - 819200 a b c - 819200 a b c 3 3 9 3 5 9 5 3 9 + 13107200 a b c - 55050240 a b c - 55050240 a b c 5 5 9 5 9 5 9 + 231211008 a b c + 3440640 a b c + 3440640 a b c 5 7 9 7 5 9 7 7 9 - 352321536 a b c - 352321536 a b c + 536870912 a b c 7 3 9 7 9 3 7 9 + 83886080 a b c - 5242880 a b c + 83886080 a b c 7 9 7 9 7 7 9 9 - 5242880 a b c + 536870912 a b c - 268435456 a b c 7 9 5 7 9 3 7 9 - 352321536 a b c + 83886080 a b c - 5242880 a b c 9 7 7 9 7 9 9 7 5 + 536870912 a b c - 268435456 a b c - 352321536 a b c 9 7 3 9 7 9 9 7 + 83886080 a b c - 5242880 a b c - 268435456 a b c 9 9 9 9 9 5 9 9 3 + 134217728 a b c + 176160768 a b c - 41943040 a b c 9 9 9 5 7 9 5 9 + 2621440 a b c - 352321536 a b c + 176160768 a b c 9 5 5 9 5 3 9 5 + 231211008 a b c - 55050240 a b c + 3440640 a b c 9 3 7 9 3 9 9 3 5 + 83886080 a b c - 41943040 a b c - 55050240 a b c 9 3 3 9 3 9 7 9 9 + 13107200 a b c - 819200 a b c - 5242880 a b c + 2621440 a b c 9 5 9 3 9 5 9 7 + 3440640 a b c - 819200 a b c + 51200 a b c - 352321536 a b c 5 9 9 5 9 5 5 9 3 + 176160768 a b c + 231211008 a b c - 55050240 a b c 5 9 3 9 7 3 9 9 + 3440640 a b c + 83886080 a b c - 41943040 a b c 3 9 5 3 9 3 3 9 9 7 - 55050240 a b c + 13107200 a b c - 819200 a b c - 5242880 a b c 9 9 9 5 9 3 9 + 2621440 a b c + 3440640 a b c - 819200 a b c + 51200 a b c 5 5 7 5 7 5 5 7 7 - 462422016 a b c - 462422016 a b c + 704643072 a b c 5 7 3 5 7 5 3 7 + 110100480 a b c - 6881280 a b c + 110100480 a b c 5 7 7 5 5 7 5 7 - 6881280 a b c - 462422016 a b c + 704643072 a b c 7 5 3 7 5 7 7 5 + 110100480 a b c - 6881280 a b c + 704643072 a b c 7 7 7 7 7 3 7 7 - 1073741824 a b c - 167772160 a b c + 10485760 a b c 7 3 5 7 3 7 7 3 3 + 110100480 a b c - 167772160 a b c - 26214400 a b c 7 3 7 5 7 7 7 3 + 1638400 a b c - 6881280 a b c + 10485760 a b c + 1638400 a b c 7 3 5 7 3 7 5 - 102400 a b c + 110100480 a b c + 110100480 a b c 3 7 7 3 7 3 3 7 - 167772160 a b c - 26214400 a b c + 1638400 a b c 3 3 7 3 7 5 7 7 5 - 26214400 a b c + 1638400 a b c - 6881280 a b c - 6881280 a b c 7 7 7 3 7 3 7 + 10485760 a b c + 1638400 a b c - 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120946279055360 a b c 9 15 11 9 15 9 - 589613110394880 a b c + 415752834252800 a b c 9 15 7 9 15 5 - 155907312844800 a b c + 29764123361280 a b c 9 15 3 9 15 - 2480343613440 a b c + 59055800320 a b c 11 15 13 11 15 15 - 600333348765696 a b c + 171523813933056 a b c 11 15 11 11 15 9 + 836178592923648 a b c - 589613110394880 a b c 11 15 7 11 15 5 + 221104916398080 a b c - 42210938585088 a b c 11 15 3 11 15 + 3517578215424 a b c - 83751862272 a b c 13 15 13 13 15 15 + 431008558088192 a b c - 123145302310912 a b c 13 15 11 13 15 9 - 600333348765696 a b c + 423311976693760 a b c 13 15 7 13 15 5 - 158741991260160 a b c + 30305289240576 a b c 13 15 3 13 15 - 2525440770048 a b c + 60129542144 a b c 15 11 13 15 11 15 - 600333348765696 a b c + 171523813933056 a b c 15 11 11 15 11 9 + 836178592923648 a b c - 589613110394880 a b c 15 11 7 15 11 5 + 221104916398080 a b c - 42210938585088 a b c 15 11 3 15 11 + 3517578215424 a b c - 83751862272 a b c 15 13 13 15 13 15 + 431008558088192 a b c - 123145302310912 a b c 15 13 11 15 13 9 - 600333348765696 a b c + 423311976693760 a b c 15 13 7 15 13 5 - 158741991260160 a b c + 30305289240576 a b c 15 13 3 15 13 - 2525440770048 a b c + 60129542144 a b c 15 15 13 15 15 15 - 123145302310912 a b c + 35184372088832 a b c 15 15 11 15 15 9 + 171523813933056 a b c - 120946279055360 a b c 15 15 7 15 15 5 + 45354854645760 a b c - 8658654068736 a b c 15 15 3 15 15 + 721554505728 a b c - 17179869184 a b c QED! This took, 713.983, seconds .