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}, {[1, 1], [2, 2, 2]}], (s^2-3)/(s^3+s^2-1), []] [[{1, 2}, {[1, 1], [2, 2, 2, 2]}], (s^3+2*s^2-4)/(s+1)/(s^3+s-1), [308, 1155, 49196]] [[{1, 2}, {[1, 1], [2, 2, 2, 2, 2]}], (s^4+2*s^3+3*s^2-5)/(s^5+s^4+s^3+s^2-1), [36, 40, 72, 289, 1404, 2808, 8964, 26244, 44712, 52488]] [[{1, 2}, {[1, 1], [2, 2, 2, 2, 2, 2]}], (s^5+2*s^4+3*s^3+4*s^2-6)/(s+1)/(s^5+s ^3+s-1), [320, 10240, 13185, 16129, 17582]] [[{1, 2}, {[1, 1], [2, 2, 2, 2, 2, 2, 2]}], (s^6+2*s^5+3*s^4+4*s^3+5*s^2-7)/(s^ 7+s^6+s^5+s^4+s^3+s^2-1), [36, 102, 6707, 9287, 68126, 93234]] [[{1, 2}, {[1, 1, 1], [2, 2, 2, 2]}], (2*s^4+4*s^3+3*s^2-5)/(s^5+2*s^4+2*s^3+s^ 2-1), [49, 143, 190, 343, 836, 3721, 40670]] [[{1, 2}, {[1, 1, 1], [2, 2, 2, 2, 2]}], (2*s^5+4*s^4+6*s^3+4*s^2-6)/(s^6+2*s^5 +2*s^4+2*s^3+s^2-1), [20, 636, 1062, 1134, 1470, 1582, 2146, 11367, 21546, 24398, 25330]] [[{1, 2}, {[1, 1, 1], [2, 2, 2, 2, 2, 2]}], (2*s^6+4*s^5+6*s^4+8*s^3+5*s^2-7)/( s^2+s+1)/(s^5+s^4+s^2+s-1), [289, 4428, 18784, 19402, 21744]] [[{1, 2}, {[1, 1, 1], [2, 2, 2, 2, 2, 2, 2]}], (2*s^7+4*s^6+6*s^5+8*s^4+10*s^3+ 6*s^2-8)/(s^8+2*s^7+2*s^6+2*s^5+2*s^4+2*s^3+s^2-1), [20, 114, 187, 333, 351, 360, 1030, 1800, 2001, 35049, 57558, 87480, 93717]] [[{1, 2}, {[1, 1, 1, 1], [2, 2, 2, 2, 2]}], (2*s^6+6*s^5+9*s^4+8*s^3+5*s^2-7)/( s^7+2*s^6+3*s^5+3*s^4+2*s^3+s^2-1), [25, 28, 66, 125, 3721, 6507, 25046, 52133] ] [[{1, 2}, {[1, 1, 1, 1], [2, 2, 2, 2, 2, 2]}], (2*s^7+6*s^6+9*s^5+12*s^4+10*s^3 +6*s^2-8)/(s+1)/(s^7+s^6+2*s^5+s^4+2*s^3+s-1), [25, 121, 169, 196, 1104, 13836] ] [[{1, 2}, {[1, 1, 1, 1], [2, 2, 2, 2, 2, 2, 2]}], (2*s^8+6*s^7+9*s^6+12*s^5+15* s^4+12*s^3+7*s^2-9)/(s^9+2*s^8+3*s^7+3*s^6+3*s^5+3*s^4+2*s^3+s^2-1), [25, 49, 125, 143, 625, 3125, 14573, 14772, 15625]] [[{1, 2}, {[1, 1, 1, 1, 1], [2, 2, 2, 2, 2, 2]}], (2*s^8+6*s^7+12*s^6+16*s^5+15 *s^4+12*s^3+7*s^2-9)/(s^9+2*s^8+3*s^7+4*s^6+4*s^5+3*s^4+2*s^3+s^2-1), [529, 16669, 53399]] [[{1, 2}, {[1, 1, 1, 1, 1], [2, 2, 2, 2, 2, 2, 2]}], (2*s^9+6*s^8+12*s^7+16*s^6 +20*s^5+18*s^4+14*s^3+8*s^2-10)/(s^10+2*s^9+3*s^8+4*s^7+4*s^6+4*s^5+3*s^4+2*s^3 +s^2-1), [28, 154, 418, 792, 803, 1040, 1525, 1918, 13113, 76272]] [[{1, 2}, {[1, 1, 1, 1, 1, 1], [2, 2, 2, 2, 2, 2, 2]}], (2*s^10+6*s^9+12*s^8+20 *s^7+25*s^6+24*s^5+21*s^4+16*s^3+9*s^2-11)/(s^11+2*s^10+3*s^9+4*s^8+5*s^7+5*s^6 +4*s^5+3*s^4+2*s^3+s^2-1), [969, 1391, 1752, 1974, 3596, 5041, 6119, 43751]] [[{1, 2, 3}, {[1, 1], [2, 2, 2]}], (3*s^3+6*s^2+3*s-4)/(s^4+3*s^3+3*s^2+s-1), [ 6, 9, 25, 4097]] [[{1, 2, 3}, {[1, 1], [2, 2, 2, 2]}], (3*s^4+6*s^3+9*s^2+4*s-5)/(s+1)/(s^4+2*s^ 3+s^2+2*s-1), [6, 9, 12, 132, 708, 1053, 1073, 3771, 16627, 38809, 82308]] [[{1, 2, 3}, {[1, 1], [2, 2, 2, 2, 2]}], (3*s^5+6*s^4+9*s^3+12*s^2+5*s-6)/(s^6+ 3*s^5+3*s^4+3*s^3+3*s^2+s-1), [6, 9, 25, 125, 370, 6241, 17854, 63650]] [[{1, 2, 3}, {[1, 1], [2, 2, 2, 2, 2, 2]}], (3*s^6+6*s^5+9*s^4+12*s^3+15*s^2+6* s-7)/(s+1)/(s^6+2*s^5+s^4+2*s^3+s^2+2*s-1), [6, 9, 18, 25, 125, 378, 1225, 3402 , 10206, 30618]] [[{1, 2, 3}, {[1, 1], [2, 2, 2, 2, 2, 2, 2]}], (3*s^7+6*s^6+9*s^5+12*s^4+15*s^3 +18*s^2+7*s-8)/(s^8+3*s^7+3*s^6+3*s^5+3*s^4+3*s^3+3*s^2+s-1), [6, 9, 25, 12054, 13390, 39314, 52441]] [[{1, 2, 3}, {[1, 1, 1], [2, 2, 2, 2]}], (3*s^5+10*s^4+15*s^3+12*s^2+5*s-6)/(s^ 6+3*s^5+5*s^4+5*s^3+3*s^2+s-1), [6, 25, 121, 169, 56454]] [[{1, 2, 3}, {[1, 1, 1], [2, 2, 2, 2, 2]}], (3*s^6+10*s^5+15*s^4+20*s^3+15*s^2+ 6*s-7)/(s^7+3*s^6+5*s^5+5*s^4+5*s^3+3*s^2+s-1), [6, 49, 1937, 27274, 59150]] [[{1, 2, 3}, {[1, 1, 1], [2, 2, 2, 2, 2, 2]}], (3*s^7+10*s^6+15*s^5+20*s^4+25*s ^3+18*s^2+7*s-8)/(s^2+s+1)/(s^6+2*s^5+2*s^4+s^3+2*s^2+2*s-1), [6, 170, 618, 1210, 6050, 12769, 30250, 42350, 52546]] [[{1, 2, 3}, {[1, 1, 1], [2, 2, 2, 2, 2, 2, 2]}], (3*s^8+10*s^7+15*s^6+20*s^5+ 25*s^4+30*s^3+21*s^2+8*s-9)/(s^9+3*s^8+5*s^7+5*s^6+5*s^5+5*s^4+5*s^3+3*s^2+s-1) , [6, 20, 60, 925, 1640, 4280, 34660, 51022, 85146]] [[{1, 2, 3}, {[1, 1, 1, 1], [2, 2, 2, 2, 2]}], (3*s^7+10*s^6+21*s^5+28*s^4+25*s ^3+18*s^2+7*s-8)/(s^8+3*s^7+5*s^6+7*s^5+7*s^4+5*s^3+3*s^2+s-1), [6, 63, 189, 238, 678, 775, 3685]] [[{1, 2, 3}, {[1, 1, 1, 1], [2, 2, 2, 2, 2, 2]}], (3*s^8+10*s^7+21*s^6+28*s^5+ 35*s^4+30*s^3+21*s^2+8*s-9)/(s+1)/(s^8+2*s^7+3*s^6+4*s^5+3*s^4+4*s^3+s^2+2*s-1) , [6, 30, 511, 961, 1140, 4371, 19311, 64180]] [[{1, 2, 3}, {[1, 1, 1, 1], [2, 2, 2, 2, 2, 2, 2]}], (3*s^9+10*s^8+21*s^7+28*s^ 6+35*s^5+42*s^4+35*s^3+24*s^2+9*s-10)/(s^10+3*s^9+5*s^8+7*s^7+7*s^6+7*s^5+7*s^4 +5*s^3+3*s^2+s-1), [6, 49, 70, 4489, 6813, 21655, 54023, 88777]] [[{1, 2, 3}, {[1, 1, 1, 1, 1], [2, 2, 2, 2, 2, 2]}], (3*s^9+10*s^8+21*s^7+36*s^ 6+45*s^5+42*s^4+35*s^3+24*s^2+9*s-10)/(s^10+3*s^9+5*s^8+7*s^7+9*s^6+9*s^5+7*s^4 +5*s^3+3*s^2+s-1), [6, 133, 738, 8138, 65302, 71003]] [[{1, 2, 3}, {[1, 1, 1, 1, 1], [2, 2, 2, 2, 2, 2, 2]}], (3*s^10+10*s^9+21*s^8+ 36*s^7+45*s^6+54*s^5+49*s^4+40*s^3+27*s^2+10*s-11)/(s^11+3*s^10+5*s^9+7*s^8+9*s ^7+9*s^6+9*s^5+7*s^4+5*s^3+3*s^2+s-1), [6, 99, 102, 289, 294, 1849, 2442, 7070] ] [[{1, 2, 3}, {[1, 1, 1, 1, 1, 1], [2, 2, 2, 2, 2, 2, 2]}], (3*s^11+10*s^10+21*s ^9+36*s^8+55*s^7+66*s^6+63*s^5+56*s^4+45*s^3+30*s^2+11*s-12)/(s^12+3*s^11+5*s^ 10+7*s^9+9*s^8+11*s^7+11*s^6+9*s^5+7*s^4+5*s^3+3*s^2+s-1), [6, 90, 330, 1309, 10719, 11453]] 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], [2, 2, 2]}], (s^2-3)/(s^3+s^2-1), []] --------------------- This took, 4507.225, seconds.