--------------------------------------- All The , 10, -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,, 10, -continued fraction judging by the first, 2000, terms [5, 11, 13, 17, 19, 23, 31, 43, 47, 61, 71, 73, 79, 97, 103, 107, 109, 113, 127 , 137, 139, 151, 157, 163, 181, 191, 193, 199, 233, 239, 241, 251, 269, 271, 277, 283, 313, 317, 331, 337, 347, 349, 353, 367, 379, 383, 389, 397, 409, 419, 421, 431, 433, 449, 457, 463, 467, 479, 487, 491, 499, 503, 521, 523, 541, 547, 557, 563, 571, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 683, 691, 709, 719, 743, 751, 757, 761, 769, 773, 787, 797, 811, 823, 827, 829, 853, 857, 859, 863, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 977, 983, 991, 997, 1009, 1013, 1019, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1097, 1103, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217] Among those that do, the largest period is the square-root of, 809, whose period is, 14 Here are all the successful ones 2, : , [[1, 24], [70, 28]] 3, : , [[1, 13], [15, 68, 15, 16]] 5, : , FAIL 7, : , [[2, 15], [20, 17, 65, 17]] 11, : , FAIL 13, : , FAIL 17, : , FAIL 19, : , FAIL 23, : , FAIL 29, : , [[5], [25, 10]] 31, : , FAIL 37, : , [[6], [120, 12]] 41, : , [[6], [24, 12]] 43, : , FAIL 47, : , FAIL 53, : , [[7], [35, 14]] 59, : , [[7], [14]] 61, : , FAIL 67, : , [[8], [53, 10, 19, 20, 13, 64, 13, 20, 19, 10, 53, 16]] 71, : , FAIL 73, : , FAIL 79, : , FAIL 83, : , [[9], [90, 18]] 89, : , [[9], [23, 235, 23, 18]] 97, : , FAIL 101, : , [[10], [200, 20]] 103, : , FAIL 107, : , FAIL 109, : , FAIL 113, : , FAIL 127, : , FAIL 131, : , [[11], [22]] 137, : , FAIL 139, : , FAIL 149, : , [[12], [48, 24]] 151, : , FAIL 157, : , FAIL 163, : , FAIL 167, : , [[12], [10, 11, 10, 24]] 173, : , [[13], [65, 26]] 179, : , [[13], [26]] 181, : , FAIL 191, : , FAIL 193, : , FAIL 197, : , [[14], [280, 28]] 199, : , FAIL 211, : , [[14], [19, 580, 19, 28]] 223, : , [[14], [10, 13, 10, 28]] 227, : , [[15], [150, 30]] 229, : , [[15], [75, 30]] 233, : , FAIL 239, : , FAIL 241, : , FAIL 251, : , FAIL 257, : , [[16], [320, 32]] 263, : , [[16], [46, 405, 46, 32]] 269, : , FAIL 271, : , FAIL 277, : , FAIL 281, : , [[16], [13, 95, 268, 95, 13, 32]] 283, : , FAIL 293, : , [[17], [85, 34]] 307, : , [[17], [19, 56, 4380, 56, 19, 34]] 311, : , [[17], [15, 13, 22, 440, 22, 13, 15, 34]] 313, : , FAIL 317, : , FAIL 331, : , FAIL 337, : , FAIL 347, : , FAIL 349, : , FAIL 353, : , FAIL 359, : , [[18], [10, 17, 10, 36]] 367, : , FAIL 373, : , [[19], [31, 10, 12, 12, 95, 12, 12, 10, 31, 38]] 379, : , FAIL 383, : , FAIL 389, : , FAIL 397, : , FAIL 401, : , [[20], [400, 40]] 409, : , FAIL 419, : , FAIL 421, : , FAIL 431, : , FAIL 433, : , FAIL 439, : , [[20], [10, 19, 10, 40]] 443, : , [[21], [210, 42]] 449, : , FAIL 457, : , FAIL 461, : , [[21], [21, 42]] 463, : , FAIL 467, : , FAIL 479, : , FAIL 487, : , FAIL 491, : , FAIL 499, : , FAIL 503, : , FAIL 509, : , [[22], [17, 12, 77, 5640, 77, 12, 17, 44]] 521, : , FAIL 523, : , FAIL 541, : , FAIL 547, : , FAIL 557, : , FAIL 563, : , FAIL 569, : , [[23], [11, 14, 595, 14, 11, 46]] 571, : , FAIL 577, : , [[24], [480, 48]] 587, : , FAIL 593, : , FAIL 599, : , FAIL 601, : , FAIL 607, : , FAIL 613, : , FAIL 617, : , FAIL 619, : , FAIL 631, : , FAIL 641, : , FAIL 643, : , FAIL 647, : , FAIL 653, : , FAIL 659, : , FAIL 661, : , FAIL 673, : , FAIL 677, : , [[26], [520, 52]] 683, : , FAIL 691, : , FAIL 701, : , [[26], [20, 10, 104, 10, 20, 52]] 709, : , FAIL 719, : , FAIL 727, : , [[26], [10, 25, 10, 52]] 733, : , [[27], [135, 54]] 739, : , [[27], [54]] 743, : , FAIL 751, : , FAIL 757, : , FAIL 761, : , FAIL 769, : , FAIL 773, : , FAIL 787, : , FAIL 797, : , FAIL 809, : , [[28], [22, 17, 30, 14, 15, 18, 710, 18, 15, 14, 30, 17, 22, 56]] 811, : , FAIL 821, : , [[28], [15, 32, 114, 32, 15, 56]] 823, : , FAIL 827, : , FAIL 829, : , FAIL 839, : , [[28], [10, 27, 10, 56]] 853, : , FAIL 857, : , FAIL 859, : , FAIL 863, : , FAIL 877, : , [[29], [16, 35, 20, 70, 592, 70, 20, 35, 16, 58]] 881, : , [[29], [14, 14, 10, 10, 10, 14, 14, 58]] 883, : , FAIL 887, : , FAIL 907, : , FAIL 911, : , FAIL 919, : , FAIL 929, : , FAIL 937, : , FAIL 941, : , FAIL 947, : , FAIL 953, : , FAIL 967, : , FAIL 971, : , [[31], [62]] 977, : , FAIL 983, : , FAIL 991, : , FAIL 997, : , FAIL 1009, : , FAIL 1013, : , FAIL 1019, : , FAIL 1021, : , [[31], [10, 20, 31, 20, 10, 62]] 1031, : , FAIL 1033, : , FAIL 1039, : , FAIL 1049, : , FAIL 1051, : , FAIL 1061, : , FAIL 1063, : , FAIL 1069, : , FAIL 1087, : , [[32], [10, 31, 10, 64]] 1091, : , [[33], [330, 66]] 1093, : , [[33], [165, 66]] 1097, : , FAIL 1103, : , FAIL 1109, : , [[33], [33, 66]] 1117, : , FAIL 1123, : , FAIL 1129, : , FAIL 1151, : , FAIL 1153, : , FAIL 1163, : , FAIL 1171, : , FAIL 1181, : , FAIL 1187, : , FAIL 1193, : , FAIL 1201, : , FAIL 1213, : , FAIL 1217, : , FAIL 1223, : , [[34], [10, 33, 10, 68]] Finally, the sequence of periods (for the OEIS) is [2, 4, FAIL, 4, FAIL, FAIL, FAIL, FAIL, FAIL, 2, FAIL, 2, 2, FAIL, FAIL, 2, 1, FAIL, 12, FAIL, FAIL, FAIL, 2, 4, FAIL, 2, FAIL, FAIL, FAIL, FAIL, FAIL, 1, FAIL, FAIL, 2, FAIL, FAIL, FAIL, 4, 2, 1, FAIL, FAIL, FAIL, 2, FAIL, 4, 4, 2, 2 , FAIL, FAIL, FAIL, FAIL, 2, 4, FAIL, FAIL, FAIL, 6, FAIL, 2, 6, 8, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 4, FAIL, 10, FAIL, FAIL, FAIL, FAIL, 2, FAIL, FAIL, FAIL, FAIL, FAIL, 4, 2, FAIL, FAIL, 2, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL , FAIL, 8, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 6, FAIL, 2, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 2 , FAIL, FAIL, 6, FAIL, FAIL, 4, 2, 1, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 14, FAIL, 6, FAIL, FAIL, FAIL, 4, FAIL, FAIL, FAIL, FAIL, 10, 8, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 1, FAIL, FAIL, FAIL , FAIL, FAIL, FAIL, FAIL, 6, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 4, 2, 2, FAIL, FAIL, 2, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL , FAIL, FAIL, FAIL, 4] This ends this article, that took, 597.681, seconds to generate.