--------------------------------------- All The , 6, -continued fractions of the first, 200, primes By Shalosh B. Ekhad The square-roots of the following primes do not seem to have an umtimately \ periodic,, 6, -continued fraction judging by the first, 2000, terms [5, 23, 43, 59, 89, 107, 113, 131, 137, 149, 151, 163, 193, 211, 233, 239, 241, 263, 281, 307, 311, 317, 331, 337, 353, 379, 383, 389, 409, 419, 421, 431, 449, 457, 463, 467, 479, 491, 499, 509, 521, 523, 563, 571, 587, 593, 613, 617, 619, 641, 643, 653, 659, 683, 691, 701, 709, 719, 739, 743, 751, 757, 773, 797, 809, 811, 821, 827, 829, 877, 881, 887, 907, 919, 929, 937, 941, 947, 953, 971, 977, 997, 1021, 1031, 1033, 1049, 1051, 1061, 1069, 1097, 1103, 1109, 1117, 1123, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1217] Among those that do, the largest period is the square-root of, 1019, whose period is, 36 Here are all the successful ones 2, : , [[1, 14], [12, 16]] 3, : , [[1, 8], [30, 10]] 5, : , FAIL 7, : , [[2, 9], [20, 10]] 11, : , [[3], [18, 6]] 13, : , [[3], [9, 6]] 17, : , [[4], [48, 8]] 19, : , [[4], [16, 8]] 23, : , FAIL 29, : , [[5], [15, 10]] 31, : , [[5], [10]] 37, : , [[6], [72, 12]] 41, : , [[6], [14, 6, 7, 9, 6, 27, 6, 9, 7, 6, 14, 12]] 43, : , FAIL 47, : , [[6], [7, 492, 7, 12]] 53, : , [[7], [21, 14]] 59, : , FAIL 61, : , [[7], [7, 14]] 67, : , [[8], [32, 16]] 71, : , [[8], [14, 75, 14, 16]] 73, : , [[8], [11, 204, 11, 16]] 79, : , [[8], [6, 7, 6, 16]] 83, : , [[9], [54, 18]] 89, : , FAIL 97, : , [[9], [7, 87, 7, 18]] 101, : , [[10], [120, 20]] 103, : , [[10], [40, 20]] 107, : , FAIL 109, : , [[10], [13, 9, 10, 11, 6, 26, 6, 11, 10, 9, 13, 20]] 113, : , FAIL 127, : , [[11], [22]] 131, : , FAIL 137, : , FAIL 139, : , [[11], [7, 10, 105, 10, 7, 22]] 149, : , FAIL 151, : , FAIL 157, : , [[12], [11, 18, 9, 81, 16, 33, 150, 33, 16, 81, 9, 18, 11, 24]] 163, : , FAIL 167, : , [[12], [6, 11, 6, 24]] 173, : , [[13], [39, 26]] 179, : , [[13], [15, 7, 23, 6, 7, 6, 23, 7, 15, 26]] 181, : , [[13], [13, 26]] 191, : , [[13], [7, 19, 84, 6, 84, 19, 7, 26]] 193, : , FAIL 197, : , [[14], [168, 28]] 199, : , [[14], [56, 28]] 211, : , FAIL 223, : , [[14], [6, 13, 6, 28]] 227, : , [[15], [90, 30]] 229, : , [[15], [45, 30]] 233, : , FAIL 239, : , FAIL 241, : , FAIL 251, : , [[15], [7, 51, 380, 51, 7, 30]] 257, : , [[16], [192, 32]] 263, : , FAIL 269, : , [[16], [14, 6, 20, 6, 14, 32]] 271, : , [[16], [12, 6, 64, 6, 12, 32]] 277, : , [[16], [9, 18, 16, 18, 9, 32]] 281, : , FAIL 283, : , [[16], [7, 20, 14, 12, 14, 20, 7, 32]] 293, : , [[17], [51, 34]] 307, : , FAIL 311, : , FAIL 313, : , [[17], [8, 8, 6, 10, 6, 8, 8, 34]] 317, : , FAIL 331, : , FAIL 337, : , FAIL 347, : , [[18], [9, 10, 7, 14, 21, 8, 11, 148, 11, 8, 21, 14, 7, 10, 9, 36]] 349, : , [[18], [8, 7, 12, 7, 2016, 7, 12, 7, 8, 36]] 353, : , FAIL 359, : , [[18], [6, 17, 6, 36]] 367, : , [[19], [38]] 373, : , [[19], [19, 38]] 379, : , FAIL 383, : , FAIL 389, : , FAIL 397, : , [[19], [6, 12, 19, 12, 6, 38]] 401, : , [[20], [240, 40]] 409, : , FAIL 419, : , FAIL 421, : , FAIL 431, : , FAIL 433, : , [[20], [7, 14, 20, 14, 7, 40]] 439, : , [[20], [6, 19, 6, 40]] 443, : , [[21], [126, 42]] 449, : , FAIL 457, : , FAIL 461, : , [[21], [12, 8, 63, 8, 12, 42]] 463, : , FAIL 467, : , FAIL 479, : , FAIL 487, : , [[22], [88, 44]] 491, : , FAIL 499, : , FAIL 503, : , [[22], [14, 201, 14, 44]] 509, : , FAIL 521, : , FAIL 523, : , FAIL 541, : , [[23], [23, 46]] 547, : , [[23], [15, 12, 6, 30, 23, 30, 6, 12, 15, 46]] 557, : , [[23], [9, 6, 69, 6, 9, 46]] 563, : , FAIL 569, : , [[23], [7, 213, 7, 46]] 571, : , FAIL 577, : , [[24], [288, 48]] 587, : , FAIL 593, : , FAIL 599, : , [[24], [12, 9, 20, 17, 18, 43, 8, 23, 8, 43, 18, 17, 20, 9, 12, 48]] 601, : , [[24], [11, 9, 18, 9, 89, 6, 6, 6, 25, 21, 7, 28, 25, 6, 16, 65, 294, 65, 16, 6, 25, 28, 7, 21, 25, 6, 6, 6, 89, 9, 18, 9, 11, 48]] 607, : , [[24], [9, 14, 11, 6, 10, 196, 10, 6, 11, 14, 9, 48]] 613, : , FAIL 617, : , FAIL 619, : , FAIL 631, : , [[25], [50]] 641, : , FAIL 643, : , FAIL 647, : , [[25], [13, 7, 6, 16, 8, 24, 8, 16, 6, 7, 13, 50]] 653, : , FAIL 659, : , FAIL 661, : , [[25], [8, 13, 21, 68, 21, 13, 8, 50]] 673, : , [[25], [6, 16, 19, 58, 99, 6, 24, 233, 24, 6, 99, 58, 19, 16, 6, 50]] 677, : , [[26], [312, 52]] 683, : , FAIL 691, : , FAIL 701, : , FAIL 709, : , FAIL 719, : , FAIL 727, : , [[26], [6, 25, 6, 52]] 733, : , [[27], [81, 54]] 739, : , FAIL 743, : , FAIL 751, : , FAIL 757, : , FAIL 761, : , [[27], [10, 25, 11, 63, 6, 4467, 6, 63, 11, 25, 10, 54]] 769, : , [[27], [8, 28, 9, 9, 18, 62, 83, 62, 18, 9, 9, 28, 8, 54]] 773, : , FAIL 787, : , [[28], [112, 56]] 797, : , FAIL 809, : , FAIL 811, : , FAIL 821, : , FAIL 823, : , [[28], [8, 8, 18, 7, 11, 6, 13, 14, 27, 12, 78, 6, 20, 67, 87, 10, 257, 10, 87, 67, 20, 6, 78, 12, 27, 14, 13, 6, 11, 7, 18, 8, 8, 56]] 827, : , FAIL 829, : , FAIL 839, : , [[28], [6, 27, 6, 56]] 853, : , [[29], [29, 58]] 857, : , [[29], [21, 7, 174, 7, 21, 58]] 859, : , [[29], [19, 13, 7, 6, 22, 107, 6, 20, 6, 107, 22, 6, 7, 13, 19, 58]] 863, : , [[29], [15, 6, 11, 9, 30, 264, 30, 9, 11, 6, 15, 58]] 877, : , FAIL 881, : , FAIL 883, : , [[29], [8, 15, 12, 7, 22, 13, 27, 44, 9, 801, 9, 44, 27, 13, 22, 7, 12, 15, 8, 58] ] 887, : , FAIL 907, : , FAIL 911, : , [[30], [32, 7, 23, 12, 29, 41, 10, 29, 10, 41, 29, 12, 23, 7, 32, 60]] 919, : , FAIL 929, : , FAIL 937, : , FAIL 941, : , FAIL 947, : , FAIL 953, : , FAIL 967, : , [[31], [62]] 971, : , FAIL 977, : , FAIL 983, : , [[31], [17, 1128, 17, 62]] 991, : , [[31], [12, 12, 62]] 997, : , FAIL 1009, : , [[31], [7, 7, 62]] 1013, : , [[31], [7, 24, 95, 24, 7, 62]] 1019, : , [[31], [6, 11, 7, 9, 25, 16, 7, 6, 9, 20, 27, 13, 11, 24, 8, 21, 6, 34, 6, 21, 8, 24, 11, 13, 27, 20, 9, 6, 7, 16, 25, 9, 7, 11, 6, 62]] 1021, : , FAIL 1031, : , FAIL 1033, : , FAIL 1039, : , [[32], [25, 8, 9, 31, 9, 8, 25, 64]] 1049, : , FAIL 1051, : , FAIL 1061, : , FAIL 1063, : , [[32], [9, 6, 15, 14, 6, 28, 130, 28, 6, 14, 15, 6, 9, 64]] 1069, : , FAIL 1087, : , [[32], [6, 31, 6, 64]] 1091, : , [[33], [198, 66]] 1093, : , [[33], [99, 66]] 1097, : , FAIL 1103, : , FAIL 1109, : , FAIL 1117, : , FAIL 1123, : , FAIL 1129, : , [[33], [9, 6, 99, 6, 9, 66]] 1151, : , FAIL 1153, : , FAIL 1163, : , FAIL 1171, : , FAIL 1181, : , FAIL 1187, : , FAIL 1193, : , FAIL 1201, : , FAIL 1213, : , [[34], [7, 24, 12, 14, 34, 14, 12, 24, 7, 68]] 1217, : , FAIL 1223, : , [[34], [6, 33, 6, 68]] Finally, the sequence of periods (for the OEIS) is [2, 2, FAIL, 2, 2, 2, 2, 2, FAIL, 2, 1, 2, 12, FAIL, 4, 2, FAIL, 2, 2, 4, 4, 4, 2, FAIL, 4, 2, 2, FAIL, 12, FAIL, 1, FAIL, FAIL, 6, FAIL, FAIL, 14, FAIL, 4, 2, 10, 2, 8, FAIL, 2, 2, FAIL, 4, 2, 2, FAIL, FAIL, FAIL, 6, 2, FAIL, 6, 6, 6, FAIL, 8, 2, FAIL, FAIL, 8, FAIL, FAIL, FAIL, 16, 10, FAIL, 4, 1, 2, FAIL, FAIL, FAIL, 6, 2, FAIL, FAIL, FAIL, FAIL, 6, 4, 2, FAIL, FAIL, 6, FAIL, FAIL, FAIL, 2 , FAIL, FAIL, 4, FAIL, FAIL, FAIL, 2, 10, 6, FAIL, 4, FAIL, 2, FAIL, FAIL, 16, 34, 12, FAIL, FAIL, FAIL, 1, FAIL, FAIL, 12, FAIL, FAIL, 8, 16, 2, FAIL, FAIL, FAIL, FAIL, FAIL, 4, 2, FAIL, FAIL, FAIL, FAIL, 12, 14, FAIL, 2, FAIL, FAIL, FAIL, FAIL, 34, FAIL, FAIL, 4, 2, 6, 16, 12, FAIL, FAIL, 20, FAIL, FAIL, 16, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 1, FAIL, FAIL, 4, 3, FAIL, 3, 6, 36, FAIL, FAIL, FAIL, 8, FAIL, FAIL, FAIL, 14, FAIL, 4, 2, 2, FAIL, FAIL, FAIL, FAIL, FAIL, 6, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 10, FAIL, 4] This ends this article, that took, 367.361, seconds to generate.