All The , 7, -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,, 7, -continued fraction judging by the first, 2000, terms [3, 5, 7, 11, 13, 19, 29, 31, 41, 47, 59, 61, 67, 73, 89, 109, 113, 131, 139, 149, 157, 163, 173, 179, 181, 191, 193, 199, 211, 229, 233, 241, 251, 269, 271, 277, 281, 283, 293, 311, 313, 317, 337, 347, 349, 353, 367, 373, 379, 383, 389, 397, 409, 419, 421, 431, 433, 449, 457, 461, 463, 467, 487, 499, 503, 509, 523, 541, 547, 557, 563, 569, 571, 587, 593, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 691, 701, 709, 719, 733, 739, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 853, 857, 859, 863, 877, 881, 883, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1093, 1097, 1109, 1117, 1123, 1129, 1151, 1153, 1171, 1181, 1187, 1193, 1201, 1213, 1217] Among those that do, the largest period is the square-root of, 521, whose period is, 18 Here are all the successful ones 2, : , [[1, 16], [7, 8, 7, 18]] 3, : , FAIL 5, : , FAIL 7, : , FAIL 11, : , FAIL 13, : , FAIL 17, : , [[4], [56, 8]] 19, : , FAIL 23, : , [[4], [8]] 29, : , FAIL 31, : , FAIL 37, : , [[6], [84, 12]] 41, : , FAIL 43, : , [[6], [12]] 47, : , FAIL 53, : , [[7], [24, 7, 100, 7, 24, 14]] 59, : , FAIL 61, : , FAIL 67, : , FAIL 71, : , [[8], [16]] 73, : , FAIL 79, : , [[8], [7, 7, 7, 16]] 83, : , [[9], [63, 18]] 89, : , FAIL 97, : , [[9], [8, 28, 16, 7, 136, 7, 16, 28, 8, 18]] 101, : , [[10], [140, 20]] 103, : , [[10], [47, 497, 47, 20]] 107, : , [[10], [20]] 109, : , FAIL 113, : , FAIL 127, : , [[11], [25, 7, 51, 11, 51, 7, 25, 22]] 131, : , FAIL 137, : , [[11], [9, 7, 13, 7, 162, 7, 13, 7, 9, 22]] 139, : , FAIL 149, : , FAIL 151, : , [[12], [24]] 157, : , FAIL 163, : , FAIL 167, : , [[12], [7, 11, 7, 24]] 173, : , FAIL 179, : , FAIL 181, : , FAIL 191, : , FAIL 193, : , FAIL 197, : , [[14], [196, 28]] 199, : , FAIL 211, : , FAIL 223, : , [[14], [7, 13, 7, 28]] 227, : , [[15], [105, 30]] 229, : , FAIL 233, : , FAIL 239, : , [[15], [15, 30]] 241, : , FAIL 251, : , FAIL 257, : , [[16], [224, 32]] 263, : , [[16], [32]] 269, : , FAIL 271, : , FAIL 277, : , FAIL 281, : , FAIL 283, : , FAIL 293, : , FAIL 307, : , [[17], [13, 16, 14, 8, 121, 8, 14, 16, 13, 34]] 311, : , FAIL 313, : , FAIL 317, : , FAIL 331, : , [[18], [36]] 337, : , FAIL 347, : , FAIL 349, : , FAIL 353, : , FAIL 359, : , [[18], [7, 17, 7, 36]] 367, : , FAIL 373, : , FAIL 379, : , FAIL 383, : , FAIL 389, : , FAIL 397, : , FAIL 401, : , [[20], [280, 40]] 409, : , FAIL 419, : , FAIL 421, : , FAIL 431, : , FAIL 433, : , FAIL 439, : , [[20], [7, 19, 7, 40]] 443, : , [[21], [147, 42]] 449, : , FAIL 457, : , FAIL 461, : , FAIL 463, : , FAIL 467, : , FAIL 479, : , [[21], [7, 7, 9, 1071, 9, 7, 7, 42]] 487, : , FAIL 491, : , [[22], [44]] 499, : , FAIL 503, : , FAIL 509, : , FAIL 521, : , [ [22], [8, 14, 12, 22, 14, 446, 7, 8, 318, 8, 7, 446, 14, 22, 12, 14, 8, 44] ] 523, : , FAIL 541, : , FAIL 547, : , FAIL 557, : , FAIL 563, : , FAIL 569, : , FAIL 571, : , FAIL 577, : , [[24], [336, 48]] 587, : , FAIL 593, : , FAIL 599, : , [[24], [14, 9, 23, 9, 14, 48]] 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], [364, 52]] 683, : , [[26], [52]] 691, : , FAIL 701, : , FAIL 709, : , FAIL 719, : , FAIL 727, : , [[26], [7, 25, 7, 52]] 733, : , FAIL 739, : , FAIL 743, : , [[27], [27, 54]] 751, : , FAIL 757, : , FAIL 761, : , FAIL 769, : , FAIL 773, : , FAIL 787, : , FAIL 797, : , FAIL 809, : , FAIL 811, : , FAIL 821, : , FAIL 823, : , FAIL 827, : , FAIL 829, : , FAIL 839, : , [[28], [7, 27, 7, 56]] 853, : , FAIL 857, : , FAIL 859, : , FAIL 863, : , FAIL 877, : , FAIL 881, : , FAIL 883, : , FAIL 887, : , [[29], [8, 7, 17, 29, 17, 7, 8, 58]] 907, : , [[30], [60]] 911, : , FAIL 919, : , FAIL 929, : , FAIL 937, : , FAIL 941, : , FAIL 947, : , FAIL 953, : , FAIL 967, : , FAIL 971, : , FAIL 977, : , FAIL 983, : , FAIL 991, : , FAIL 997, : , FAIL 1009, : , FAIL 1013, : , FAIL 1019, : , FAIL 1021, : , FAIL 1031, : , [[32], [64]] 1033, : , FAIL 1039, : , FAIL 1049, : , FAIL 1051, : , FAIL 1061, : , FAIL 1063, : , FAIL 1069, : , FAIL 1087, : , [[32], [7, 31, 7, 64]] 1091, : , [[33], [231, 66]] 1093, : , FAIL 1097, : , FAIL 1103, : , [[33], [33, 66]] 1109, : , FAIL 1117, : , FAIL 1123, : , FAIL 1129, : , FAIL 1151, : , FAIL 1153, : , FAIL 1163, : , [[34], [68]] 1171, : , FAIL 1181, : , FAIL 1187, : , FAIL 1193, : , FAIL 1201, : , FAIL 1213, : , FAIL 1217, : , FAIL 1223, : , [[34], [7, 33, 7, 68]] Finally, the sequence of periods (for the OEIS) is [4, FAIL, FAIL, FAIL, FAIL, FAIL, 2, FAIL, 1, FAIL, FAIL, 2, FAIL, 1, FAIL, 6, FAIL, FAIL, FAIL, 1, FAIL, 4, 2, FAIL, 10, 2, 4, 1, FAIL, FAIL, 8, FAIL, 10, FAIL, FAIL, 1, FAIL, FAIL, 4, FAIL, FAIL, FAIL, FAIL, FAIL, 2, FAIL, FAIL, 4, 2 , FAIL, FAIL, 2, FAIL, FAIL, 2, 1, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 10, FAIL , FAIL, FAIL, 1, FAIL, FAIL, FAIL, FAIL, 4, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 2, FAIL, FAIL, FAIL, FAIL, FAIL, 4, 2, FAIL, FAIL, FAIL, FAIL, FAIL, 8, FAIL, 1 , FAIL, FAIL, FAIL, 18, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 2, FAIL, FAIL , 6, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 2, 1, FAIL, FAIL, FAIL, FAIL, 4, FAIL, FAIL, 2, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 4, FAIL, FAIL, FAIL, FAIL , FAIL, FAIL, FAIL, 8, 1, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 1, FAIL, FAIL, FAIL, FAIL, FAIL , FAIL, FAIL, 4, 2, FAIL, FAIL, 2, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 1, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 4] This ends this article, that took, 654.557, seconds to generate.