All the Primality Tests Inspired by an alphabet of 2 to , 5, 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}, {[1, 1]}], (s-2)/(s^2+s-1), [705, 2465, 2737, 3745, 4181, 5777, 6721, 10877, 13201, 15251, 24465, 29281, 34561, 35785, 51841, 54705, 64079, 64681, 67861, 68251, 75077, 80189, 90061, 96049, 97921]] [[{1, 2}, {[1, 1, 1]}], (s^2+2*s-3)/(s^3+s^2+s-1), [182, 25201, 54289, 63618]] [[{1, 2}, {[1, 1, 1, 1]}], (s^3+2*s^2+3*s-4)/(s^4+s^3+s^2+s-1), [25, 49, 125, 169, 343, 493, 1369, 7825, 29877, 64253, 98093]] [[{1, 2}, {[1, 1, 1, 1, 1]}], (s^4+2*s^3+3*s^2+4*s-5)/(s^5+s^4+s^3+s^2+s-1), [ 42, 84, 121, 361, 910, 1582, 5626, 6751, 19390, 20052, 30954, 55893]] [[{1, 2}, {[1, 1, 1, 1, 1, 1]}], (s^5+2*s^4+3*s^3+4*s^2+5*s-6)/(s^6+s^5+s^4+s^3 +s^2+s-1), [186, 841, 2209, 39035, 72046]] [[{1, 2}, {[1, 1, 1, 1, 1, 1, 1]}], (s^6+2*s^5+3*s^4+4*s^3+5*s^2+6*s-7)/(s^7+s^ 6+s^5+s^4+s^3+s^2+s-1), [169, 567, 1701, 11907, 42925]] [[{1, 2, 3}, {[1, 1]}], (2*s-2)/(2*s^2+2*s-1), [6, 9, 18, 25, 27, 54, 81, 110, 138, 162, 198, 243, 486, 729, 891, 910, 1369, 1413, 1458, 1518, 1593, 1870, 2162, 2187, 2349, 2403, 2538, 2590, 4374, 5589, 5778, 6561, 8118, 8749, 10266, 11638, 12025, 13122, 13310, 13446, 16038, 17110, 17710, 19683, 19778, 20878, 22923, 23902, 24094, 24178, 27830, 29341, 33291, 33534, 35425, 39366, 39618, 40161, 41470, 46657, 54670, 58190, 59049, 63366, 67858, 72171, 79378, 88270, 88561, 89667, 93258, 99937]] [[{1, 2, 3}, {[1, 1, 1]}], (2*s^2+4*s-3)/(2*s^3+2*s^2+2*s-1), [6, 121, 245, 618 , 5041, 44834, 47878]] [[{1, 2, 3}, {[1, 1, 1, 1]}], (2*s^3+4*s^2+6*s-4)/(2*s^4+2*s^3+2*s^2+2*s-1), [6 , 49, 138, 697, 2706, 4191, 7783, 8481, 34026]] [[{1, 2, 3}, {[1, 1, 1, 1, 1]}], (2*s^4+4*s^3+6*s^2+8*s-5)/(2*s^5+2*s^4+2*s^3+2 *s^2+2*s-1), [6, 42, 102, 11513]] [[{1, 2, 3}, {[1, 1, 1, 1, 1, 1]}], (2*s^5+4*s^4+6*s^3+8*s^2+10*s-6)/(2*s^6+2*s ^5+2*s^4+2*s^3+2*s^2+2*s-1), [6, 66, 2001, 2954, 5041, 96405]] [[{1, 2, 3}, {[1, 1, 1, 1, 1, 1, 1]}], (2*s^6+4*s^5+6*s^4+8*s^3+10*s^2+12*s-7)/ (2*s^7+2*s^6+2*s^5+2*s^4+2*s^3+2*s^2+2*s-1), [6, 66, 406, 858, 29869, 83967]] [[{1, 2, 3, 4}, {[1, 1]}], (3*s-2)/(3*s^2+3*s-1), [4, 51, 65, 68, 165, 469, 867 , 1105, 1369, 5491, 7939, 8911, 40132, 50653, 62745, 65801, 75361, 83333, 88831 ]] [[{1, 2, 3, 4}, {[1, 1, 1]}], (3*s^2+6*s-3)/(3*s^3+3*s^2+3*s-1), [4, 6, 8, 12, 15, 16, 20, 24, 32, 48, 64, 75, 96, 121, 128, 192, 256, 384, 416, 512, 640, 768 , 960, 1024, 1032, 1104, 1120, 1216, 1253, 1376, 1515, 1536, 1644, 1902, 2048, 2144, 2624, 2944, 3072, 3392, 4096, 4384, 4564, 4952, 6144, 8192, 9136, 12288, 15704, 15744, 16384, 20480, 24576, 26436, 26624, 29140, 30720, 32768, 40678, 49152, 51888, 54404, 57824, 65536, 77568, 77824, 87555, 94192, 98304]] [[{1, 2, 3, 4}, {[1, 1, 1, 1]}], (3*s^3+6*s^2+9*s-4)/(3*s^4+3*s^3+3*s^2+3*s-1), [4, 6, 12, 15, 28, 308, 361, 402, 996, 1337, 2244, 22445, 26012, 39012, 44521]] [[{1, 2, 3, 4}, {[1, 1, 1, 1, 1]}], (3*s^4+6*s^3+9*s^2+12*s-5)/(3*s^5+3*s^4+3*s ^3+3*s^2+3*s-1), [4, 6, 15, 28, 42, 294, 582, 962, 1270, 1369, 4991, 5106, 8310 , 13731, 23001]] [[{1, 2, 3, 4}, {[1, 1, 1, 1, 1, 1]}], (3*s^5+6*s^4+9*s^3+12*s^2+15*s-6)/(3*s^6 +3*s^5+3*s^4+3*s^3+3*s^2+3*s-1), [4, 6, 12, 15, 66, 186, 2209, 3515, 5340, 12769, 18359, 83276]] [[{1, 2, 3, 4}, {[1, 1, 1, 1, 1, 1, 1]}], (3*s^6+6*s^5+9*s^4+12*s^3+15*s^2+18*s -7)/(3*s^7+3*s^6+3*s^5+3*s^4+3*s^3+3*s^2+3*s-1), [4, 6, 12, 15, 28, 66, 84, 186 , 2091, 5126, 13412, 87127]] [[{1, 2, 3, 4, 5}, {[1, 1]}], (4*s-2)/(4*s^2+4*s-1), [9, 10, 12, 20, 133, 655, 841, 1105, 1308, 4795, 6601, 8257, 8449, 15841, 31417, 52633, 59711, 66745, 68089, 75361]] [[{1, 2, 3, 4, 5}, {[1, 1, 1]}], (4*s^2+8*s-3)/(4*s^3+4*s^2+4*s-1), [10, 70, 490, 604, 755, 790, 1991, 8270, 9447, 17130, 24649, 86756, 88831]] [[{1, 2, 3, 4, 5}, {[1, 1, 1, 1]}], (4*s^3+8*s^2+12*s-4)/(4*s^4+4*s^3+4*s^2+4*s -1), [10, 20, 25, 49, 50, 100, 124, 125, 140, 155, 156, 170, 250, 289, 300, 476 , 500, 530, 625, 666, 750, 850, 900, 901, 1070, 1250, 1875, 2028, 2250, 2500, 2828, 2900, 3125, 3500, 3900, 4215, 4375, 6250, 6500, 8692, 11140, 11875, 12500 , 12636, 15625, 18450, 21060, 30625, 31250, 39625, 47900, 51372, 56550, 62500, 78125, 81875, 87500, 97700]] [[{1, 2, 3, 4, 5}, {[1, 1, 1, 1, 1]}], (4*s^4+8*s^3+12*s^2+16*s-5)/(4*s^5+4*s^4 +4*s^3+4*s^2+4*s-1), [10, 15, 20, 42, 65, 124, 238, 370, 3342, 18348, 21230]] [[{1, 2, 3, 4, 5}, {[1, 1, 1, 1, 1, 1]}], (4*s^5+8*s^4+12*s^3+16*s^2+20*s-6)/(4 *s^6+4*s^5+4*s^4+4*s^3+4*s^2+4*s-1), [10, 15, 20, 30, 45, 65, 124, 2060, 2370, 3721, 10230, 13988, 18260, 23340, 28086]] [[{1, 2, 3, 4, 5}, {[1, 1, 1, 1, 1, 1, 1]}], (4*s^6+8*s^5+12*s^4+16*s^3+20*s^2+ 24*s-7)/(4*s^7+4*s^6+4*s^5+4*s^4+4*s^3+4*s^2+4*s-1), [10, 15, 20, 45, 65, 66, 124, 140, 690, 5394, 10308, 11086, 13653, 98642]] The set of patterns with, followed by the generating function, with the leas\ t number of pseudoprimes less than, 100000, is: [[{1, 2}, {[1, 1, 1]}], (s^2+2*s-3)/(s^3+s^2+s-1), [182, 25201, 54289, 63618]] --------------------- This took, 228.004, seconds.