All The , 3, -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,, 3, -continued fraction

                      judging by the first, 2000,  terms



[29, 43, 53, 59, 97, 107, 113, 137, 149, 151, 157, 163, 193, 233, 239, 241, 277
, 281, 283, 293, 311, 317, 337, 353, 379, 383, 397, 409, 421, 431, 433, 449, 
457, 461, 463, 499, 509, 521, 547, 563, 569, 571, 587, 593, 601, 607, 613, 617,
619, 643, 647, 653, 659, 661, 673, 683, 701, 709, 719, 733, 739, 743, 751, 757,
761, 769, 773, 797, 809, 811, 821, 827, 829, 853, 857, 859, 863, 881, 883, 911,
929, 937, 941, 947, 953, 971, 977, 983, 997, 1009, 1013, 1021, 1031, 1033, 1039
, 1049, 1051, 1061, 1069, 1093, 1103, 1109, 1117, 1123, 1151, 1153, 1163, 1171,
1181, 1187, 1193, 1201, 1213, 1217]


 Among those that do, the largest period is the square-root of, 641,

    whose period is, 34

                        Here are all the successful ones

                            2, : , [[1, 7], [12, 8]]

                            3, : , [[1, 4], [30, 5]]

                             5, : , [[2], [12, 4]]

                               7, : , [[2], [4]]

                             11, : , [[3], [9, 6]]

                       13, : , [[3], [4, 3, 20, 3, 4, 6]]

                             17, : , [[4], [24, 8]]

                               19, : , [[4], [8]]

                          23, : , [[4], [3, 3, 3, 8]]

                                  29, : , FAIL

                             31, : , [[5], [5, 10]]

                            37, : , [[6], [36, 12]]

             41, : , [[6], [7, 6, 3, 3, 4, 114, 4, 3, 3, 6, 7, 12]]

                                  43, : , FAIL

                          47, : , [[6], [3, 5, 3, 12]]

                                  53, : , FAIL

                                  59, : , FAIL

        61, : , [[7], [3, 4, 11, 27, 3, 6, 46, 6, 3, 27, 11, 4, 3, 14]]

                              67, : , [[8], [16]]

                         71, : , [[8], [7, 75, 7, 16]]

                    73, : , [[8], [5, 5, 3, 4, 3, 5, 5, 16]]

                          79, : , [[8], [3, 7, 3, 16]]

                            83, : , [[9], [27, 18]]

89, : ,

    [[9], [6, 3, 10, 6, 5, 3, 4, 9, 3, 168, 3, 9, 4, 3, 5, 6, 10, 3, 6, 18]]

                                  97, : , FAIL

                           101, : , [[10], [60, 20]]

                             103, : , [[10], [20]]

                                 107, : , FAIL

            109, : , [[10], [6, 3, 4, 8, 9, 62, 9, 8, 4, 3, 6, 20]]

                                 113, : , FAIL

                           127, : , [[11], [11, 22]]

                     131, : , [[11], [6, 4, 33, 4, 6, 22]]

                                 137, : , FAIL

                       139, : , [[11], [3, 3, 3, 3, 22]]

                                 149, : , FAIL

                                 151, : , FAIL

                                 157, : , FAIL

                                 163, : , FAIL

                        167, : , [[12], [3, 11, 3, 24]]

               173, : , [[13], [19, 4, 3, 7, 8, 7, 3, 4, 19, 26]]

              179, : , [[13], [7, 3, 10, 5, 39, 5, 10, 3, 7, 26]]

181, : , [[13], [6, 4, 3, 8, 19, 3, 5, 11, 4, 3, 28, 12, 726, 12, 28, 3, 4, 11,

    5, 3, 19, 8, 3, 4, 6, 26]]

            191, : , [[13], [3, 4, 5, 9, 7, 13, 7, 9, 5, 4, 3, 26]]

                                 193, : , FAIL

                           197, : , [[14], [84, 28]]

                             199, : , [[14], [28]]

               211, : , [[14], [5, 4, 11, 3, 3, 3, 11, 4, 5, 28]]

                        223, : , [[14], [3, 13, 3, 28]]

                           227, : , [[15], [45, 30]]

                    229, : , [[15], [22, 5, 816, 5, 22, 30]]

                                 233, : , FAIL

                                 239, : , FAIL

                                 241, : , FAIL

               251, : , [[15], [3, 5, 8, 21, 5, 21, 8, 5, 3, 30]]

                           257, : , [[16], [96, 32]]

                 263, : , [[16], [13, 3, 4, 15, 4, 3, 13, 32]]

                     269, : , [[16], [7, 6, 10, 6, 7, 32]]

                          271, : , [[16], [6, 6, 32]]

                                 277, : , FAIL

                                 281, : , FAIL

                                 283, : , FAIL

                                 293, : , FAIL

                  307, : , [[17], [5, 3, 3, 51, 3, 3, 5, 34]]

                                 311, : , FAIL

                  313, : , [[17], [4, 8, 3, 10, 3, 8, 4, 34]]

                                 317, : , FAIL

                331, : , [[18], [15, 5, 3, 6, 6, 3, 5, 15, 36]]

                                 337, : , FAIL

               347, : , [[18], [4, 3, 3, 6, 55, 6, 3, 3, 4, 36]]

              349, : , [[18], [4, 7, 6, 7, 1008, 7, 6, 7, 4, 36]]

                                 353, : , FAIL

                        359, : , [[18], [3, 17, 3, 36]]

                           367, : , [[19], [19, 38]]

              373, : , [[19], [9, 5, 15, 3, 114, 3, 15, 5, 9, 38]]

                                 379, : , FAIL

                                 383, : , FAIL

389, : , [[19], [4, 20, 20, 8, 3, 9, 11, 3, 7, 33, 4, 5, 1064, 5, 4, 33, 7, 3,

    11, 9, 3, 8, 20, 20, 4, 38]]

                                 397, : , FAIL

                           401, : , [[20], [120, 40]]

                                 409, : , FAIL

         419, : , [[20], [6, 7, 4, 9, 10, 3, 5, 3, 10, 9, 4, 7, 6, 40]]

                                 421, : , FAIL

                                 431, : , FAIL

                                 433, : , FAIL

                        439, : , [[20], [3, 19, 3, 40]]

                           443, : , [[21], [63, 42]]

                                 449, : , FAIL

                                 457, : , FAIL

                                 461, : , FAIL

                                 463, : , FAIL

              467, : , [[21], [4, 3, 10, 3, 63, 3, 10, 3, 4, 42]]

            479, : , [[21], [3, 7, 3, 3, 6, 21, 6, 3, 3, 7, 3, 42]]

                             487, : , [[22], [44]]

       491, : , [[22], [18, 3, 12, 8, 4, 4, 597, 4, 4, 8, 12, 3, 18, 44]]

                                 499, : , FAIL

                        503, : , [[22], [7, 201, 7, 44]]

                                 509, : , FAIL

                                 521, : , FAIL

         523, : , [[22], [3, 6, 4, 4, 4, 3, 410, 3, 4, 4, 4, 6, 3, 44]]

541, : , [[23], [

    11, 5, 9, 4, 4, 6, 8, 3, 5, 3, 3, 138, 3, 3, 5, 3, 8, 6, 4, 4, 9, 5, 11, 46

    ]]

                                 547, : , FAIL

                     557, : , [[23], [4, 3, 140, 3, 4, 46]]

                                 563, : , FAIL

                                 569, : , FAIL

                                 571, : , FAIL

                           577, : , [[24], [144, 48]]

                                 587, : , FAIL

                                 593, : , FAIL

   599, : , [[24], [6, 9, 10, 17, 9, 43, 4, 23, 4, 43, 9, 17, 10, 9, 6, 48]]

                                 601, : , FAIL

                                 607, : , FAIL

                                 613, : , FAIL

                                 617, : , FAIL

                                 619, : , FAIL

                           631, : , [[25], [25, 50]]

641, : , [[25], [9, 6, 3, 9, 11, 7, 3, 6, 3, 5, 6, 5, 12, 4, 20, 3, 150, 3, 20,

    4, 12, 5, 6, 5, 3, 6, 3, 7, 11, 9, 3, 6, 9, 50]]

                                 643, : , FAIL

                                 647, : , FAIL

                                 653, : , FAIL

                                 659, : , FAIL

                                 661, : , FAIL

                                 673, : , FAIL

                           677, : , [[26], [156, 52]]

                                 683, : , FAIL

       691, : , [[26], [10, 6, 5, 8, 13, 3, 77, 3, 13, 8, 5, 6, 10, 52]]

                                 701, : , FAIL

                                 709, : , FAIL

                                 719, : , FAIL

                        727, : , [[26], [3, 25, 3, 52]]

                                 733, : , FAIL

                                 739, : , FAIL

                                 743, : , FAIL

                                 751, : , FAIL

                                 757, : , FAIL

                                 761, : , FAIL

                                 769, : , FAIL

                                 773, : , FAIL

                             787, : , [[28], [56]]

                                 797, : , FAIL

                                 809, : , FAIL

                                 811, : , FAIL

                                 821, : , FAIL

         823, : , [[28], [4, 8, 9, 7, 5, 3, 27, 3, 5, 7, 9, 8, 4, 56]]

                                 827, : , FAIL

                                 829, : , FAIL

                        839, : , [[28], [3, 27, 3, 56]]

                                 853, : , FAIL

                                 857, : , FAIL

                                 859, : , FAIL

                                 863, : , FAIL

877, : , [

    [29], [4, 3, 7, 4, 5, 6, 3, 3, 6, 3, 18, 3, 6, 3, 3, 6, 5, 4, 7, 3, 4, 58]]

                                 881, : , FAIL

                                 883, : , FAIL

                  887, : , [[29], [3, 3, 5, 267, 5, 3, 3, 58]]

907, : , [[30], [25, 3, 3, 12, 6, 6, 5, 8, 15, 6, 3, 5, 6, 3, 3, 5, 9, 5, 3, 3,

    6, 5, 3, 6, 15, 8, 5, 6, 6, 12, 3, 3, 25, 60]]

                                 911, : , FAIL

                  919, : , [[30], [9, 5, 4, 29, 4, 5, 9, 60]]

                                 929, : , FAIL

                                 937, : , FAIL

                                 941, : , FAIL

                                 947, : , FAIL

                                 953, : , FAIL

                           967, : , [[31], [31, 62]]

                                 971, : , FAIL

                                 977, : , FAIL

                                 983, : , FAIL

                    991, : , [[31], [6, 12, 31, 12, 6, 62]]

                                 997, : , FAIL

                                 1009, : , FAIL

                                 1013, : , FAIL

1019, : , [[31],

    [3, 11, 3, 3, 4, 4, 12, 5, 3, 7, 95, 7, 3, 5, 12, 4, 4, 3, 3, 11, 3, 62]]

                                 1021, : , FAIL

                                 1031, : , FAIL

                                 1033, : , FAIL

                                 1039, : , FAIL

                                 1049, : , FAIL

                                 1051, : , FAIL

                                 1061, : , FAIL

                     1063, : , [[32], [4, 3, 31, 3, 4, 64]]

                                 1069, : , FAIL

                        1087, : , [[32], [3, 31, 3, 64]]

                           1091, : , [[33], [99, 66]]

                                 1093, : , FAIL

1097, : , [[33],

    [24, 3, 3, 3, 7, 3, 5, 9, 11, 8, 198, 8, 11, 9, 5, 3, 7, 3, 3, 3, 24, 66]]

                                 1103, : , FAIL

                                 1109, : , FAIL

                                 1117, : , FAIL

                                 1123, : , FAIL

                    1129, : , [[33], [4, 3, 200, 3, 4, 66]]

                                 1151, : , FAIL

                                 1153, : , FAIL

                                 1163, : , FAIL

                                 1171, : , FAIL

                                 1181, : , FAIL

                                 1187, : , FAIL

                                 1193, : , FAIL

                                 1201, : , FAIL

                                 1213, : , FAIL

                                 1217, : , FAIL

                        1223, : , [[34], [3, 33, 3, 68]]

               Finally, the sequence of periods (for the OEIS) is

[2, 2, 2, 1, 2, 6, 2, 1, 4, FAIL, 2, 2, 12, FAIL, 4, FAIL, FAIL, 14, 1, 4, 8, 4
, 2, 20, FAIL, 2, 1, FAIL, 12, FAIL, 2, 6, FAIL, 5, FAIL, FAIL, FAIL, FAIL, 4,
10, 10, 26, 12, FAIL, 2, 1, 10, 4, 2, 6, FAIL, FAIL, FAIL, 10, 2, 8, 6, 3, FAIL
, FAIL, FAIL, FAIL, 8, FAIL, 8, FAIL, 9, FAIL, 10, 10, FAIL, 4, 2, 10, FAIL, 
FAIL, 26, FAIL, 2, FAIL, 14, FAIL, FAIL, FAIL, 4, 2, FAIL, FAIL, FAIL, FAIL, 10
, 12, 1, 14, FAIL, 4, FAIL, FAIL, 14, 24, FAIL, 6, FAIL, FAIL, FAIL, 2, FAIL, 
FAIL, 16, FAIL, FAIL, FAIL, FAIL, FAIL, 2, 34, FAIL, FAIL, FAIL, FAIL, FAIL, 
FAIL, 2, FAIL, 14, FAIL, FAIL, FAIL, 4, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 
FAIL, FAIL, 1, FAIL, FAIL, FAIL, FAIL, 14, FAIL, FAIL, 4, FAIL, FAIL, FAIL, 
FAIL, 22, FAIL, FAIL, 8, 34, FAIL, 8, FAIL, FAIL, FAIL, FAIL, FAIL, 2, FAIL, 
FAIL, FAIL, 6, FAIL, FAIL, FAIL, 22, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL,
6, FAIL, 4, 2, FAIL, 22, FAIL, FAIL, FAIL, FAIL, 6, FAIL, FAIL, FAIL, FAIL, 
FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 4]


       This ends this article, that took, 410.651, seconds to generate.