All The , 9, -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,, 9, -continued fraction judging by the first, 2000, terms [13, 17, 23, 29, 43, 47, 53, 59, 61, 79, 89, 97, 107, 113, 131, 137, 149, 151, 157, 163, 173, 179, 181, 191, 193, 211, 229, 233, 239, 241, 251, 263, 269, 281, 293, 311, 317, 331, 337, 349, 353, 373, 383, 389, 397, 419, 421, 431, 449, 457, 461, 463, 479, 491, 499, 509, 521, 523, 541, 557, 569, 571, 587, 593, 601, 607, 613, 617, 619, 641, 647, 653, 659, 661, 673, 683, 691, 701, 709, 719, 733, 739, 743, 751, 757, 761, 769, 773, 797, 809, 811, 821, 827, 829, 853, 857, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 971, 977, 983, 997, 1009 , 1013, 1019, 1021, 1031, 1039, 1049, 1051, 1061, 1069, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1181, 1187, 1193, 1213, 1217] Among those that do, the largest period is the square-root of, 599, whose period is, 16 Here are all the successful ones 2, : , [[1, 21], [12, 24]] 3, : , [[1, 12], [30, 15]] 5, : , [[2, 38], [72, 40]] 7, : , [[2, 13], [9, 14]] 11, : , [[3, 28], [21, 48, 21, 29]] 13, : , FAIL 17, : , FAIL 19, : , [[4, 25], [117, 26]] 23, : , FAIL 29, : , FAIL 31, : , [[5], [15, 10]] 37, : , [[6], [108, 12]] 41, : , [[6], [22, 27, 14, 114, 14, 27, 22, 12]] 43, : , FAIL 47, : , FAIL 53, : , FAIL 59, : , FAIL 61, : , FAIL 67, : , [[8], [48, 16]] 71, : , [[8], [21, 75, 21, 16]] 73, : , [[8], [16]] 79, : , FAIL 83, : , [[9], [81, 18]] 89, : , FAIL 97, : , FAIL 101, : , [[10], [180, 20]] 103, : , [[10], [60, 20]] 107, : , FAIL 109, : , [[10], [20]] 113, : , FAIL 127, : , [[11], [33, 22]] 131, : , FAIL 137, : , FAIL 139, : , [[11], [11, 22]] 149, : , FAIL 151, : , FAIL 157, : , FAIL 163, : , FAIL 167, : , [[12], [9, 11, 9, 24]] 173, : , FAIL 179, : , FAIL 181, : , FAIL 191, : , FAIL 193, : , FAIL 197, : , [[14], [252, 28]] 199, : , [[14], [84, 28]] 211, : , FAIL 223, : , [[14], [9, 13, 9, 28]] 227, : , [[15], [135, 30]] 229, : , FAIL 233, : , FAIL 239, : , FAIL 241, : , FAIL 251, : , FAIL 257, : , [[16], [288, 32]] 263, : , FAIL 269, : , FAIL 271, : , [[16], [19, 18, 10, 18, 19, 32]] 277, : , [[16], [13, 9, 98, 9, 13, 32]] 281, : , FAIL 283, : , [[16], [10, 9, 15, 9, 10, 32]] 293, : , FAIL 307, : , [[17], [17, 34]] 311, : , FAIL 313, : , [[17], [13, 954, 13, 34]] 317, : , FAIL 331, : , FAIL 337, : , FAIL 347, : , [[18], [14, 27, 167, 27, 14, 36]] 349, : , FAIL 353, : , FAIL 359, : , [[18], [9, 17, 9, 36]] 367, : , [[19], [57, 38]] 373, : , FAIL 379, : , [[19], [19, 38]] 383, : , FAIL 389, : , FAIL 397, : , FAIL 401, : , [[20], [360, 40]] 409, : , [[20], [40]] 419, : , FAIL 421, : , FAIL 431, : , FAIL 433, : , [[20], [11, 69, 21, 124, 21, 69, 11, 40]] 439, : , [[20], [9, 19, 9, 40]] 443, : , [[21], [189, 42]] 449, : , FAIL 457, : , FAIL 461, : , FAIL 463, : , FAIL 467, : , [[21], [14, 12, 1749, 12, 14, 42]] 479, : , FAIL 487, : , [[22], [132, 44]] 491, : , FAIL 499, : , FAIL 503, : , [[22], [21, 201, 21, 44]] 509, : , FAIL 521, : , FAIL 523, : , FAIL 541, : , FAIL 547, : , [[23], [23, 46]] 557, : , FAIL 563, : , [[23], [12, 24, 23, 24, 12, 46]] 569, : , FAIL 571, : , FAIL 577, : , [[24], [432, 48]] 587, : , FAIL 593, : , FAIL 599, : , [[24], [18, 9, 30, 17, 27, 43, 12, 23, 12, 43, 27, 17, 30, 9, 18, 48]] 601, : , FAIL 607, : , FAIL 613, : , FAIL 617, : , FAIL 619, : , FAIL 631, : , [[25], [75, 50]] 641, : , FAIL 643, : , [[25], [25, 50]] 647, : , FAIL 653, : , FAIL 659, : , FAIL 661, : , FAIL 673, : , FAIL 677, : , [[26], [468, 52]] 683, : , FAIL 691, : , FAIL 701, : , FAIL 709, : , FAIL 719, : , FAIL 727, : , [[26], [9, 25, 9, 52]] 733, : , FAIL 739, : , FAIL 743, : , FAIL 751, : , FAIL 757, : , FAIL 761, : , FAIL 769, : , FAIL 773, : , FAIL 787, : , [[28], [168, 56]] 797, : , FAIL 809, : , FAIL 811, : , FAIL 821, : , FAIL 823, : , [[28], [13, 109, 9, 31, 9, 31, 9, 109, 13, 56]] 827, : , FAIL 829, : , FAIL 839, : , [[28], [9, 27, 9, 56]] 853, : , FAIL 857, : , FAIL 859, : , [[29], [29, 58]] 863, : , FAIL 877, : , FAIL 881, : , FAIL 883, : , FAIL 887, : , FAIL 907, : , FAIL 911, : , FAIL 919, : , FAIL 929, : , FAIL 937, : , FAIL 941, : , FAIL 947, : , FAIL 953, : , FAIL 967, : , [[31], [93, 62]] 971, : , FAIL 977, : , FAIL 983, : , FAIL 991, : , [[31], [18, 12, 93, 12, 18, 62]] 997, : , FAIL 1009, : , FAIL 1013, : , FAIL 1019, : , FAIL 1021, : , FAIL 1031, : , FAIL 1033, : , [[32], [64]] 1039, : , FAIL 1049, : , FAIL 1051, : , FAIL 1061, : , FAIL 1063, : , [[32], [14, 9, 9, 9, 14, 64]] 1069, : , FAIL 1087, : , [[32], [9, 31, 9, 64]] 1091, : , [[33], [297, 66]] 1093, : , FAIL 1097, : , FAIL 1103, : , FAIL 1109, : , FAIL 1117, : , FAIL 1123, : , FAIL 1129, : , FAIL 1151, : , FAIL 1153, : , FAIL 1163, : , FAIL 1171, : , [[34], [40, 9, 13, 13, 204, 13, 13, 9, 40, 68]] 1181, : , FAIL 1187, : , FAIL 1193, : , FAIL 1201, : , [[34], [13, 12, 28, 9, 206, 9, 28, 12, 13, 68]] 1213, : , FAIL 1217, : , FAIL 1223, : , [[34], [9, 33, 9, 68]] Finally, the sequence of periods (for the OEIS) is [2, 2, 2, 2, 4, FAIL, FAIL, 2, FAIL, FAIL, 2, 2, 8, FAIL, FAIL, FAIL, FAIL, FAIL, 2, 4, 1, FAIL, 2, FAIL, FAIL, 2, 2, FAIL, 1, FAIL, 2, FAIL, FAIL, 2, FAIL , FAIL, FAIL, FAIL, 4, FAIL, FAIL, FAIL, FAIL, FAIL, 2, 2, FAIL, 4, 2, FAIL, FAIL, FAIL, FAIL, FAIL, 2, FAIL, FAIL, 6, 6, FAIL, 6, FAIL, 2, FAIL, 4, FAIL, FAIL, FAIL, 6, FAIL, FAIL, 4, 2, FAIL, 2, FAIL, FAIL, FAIL, 2, 1, FAIL, FAIL, FAIL, 8, 4, 2, FAIL, FAIL, FAIL, FAIL, 6, FAIL, 2, FAIL, FAIL, 4, FAIL, FAIL, FAIL, FAIL, 2, FAIL, 6, FAIL, FAIL, 2, FAIL, FAIL, 16, FAIL, FAIL, FAIL, FAIL, FAIL, 2, FAIL, 2, FAIL, FAIL, FAIL, FAIL, FAIL, 2, FAIL, FAIL, FAIL, FAIL, FAIL , 4, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 2, FAIL, FAIL, FAIL, FAIL, 10, FAIL, FAIL, 4, FAIL, FAIL, 2, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 2, FAIL, FAIL, FAIL, 6, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 1, FAIL, FAIL, FAIL, FAIL, 6, FAIL, 4, 2, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 10, FAIL, FAIL, FAIL, 10, FAIL, FAIL, 4] This ends this article, that took, 529.122, seconds to generate.