All The , 4, -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,, 4, -continued fraction

                      judging by the first, 2000,  terms



[19, 43, 67, 71, 73, 97, 109, 113, 127, 139, 149, 151, 157, 163, 179, 191, 211,
239, 241, 251, 271, 277, 307, 337, 349, 379, 389, 409, 419, 421, 431, 433, 467,
479, 487, 491, 503, 509, 521, 523, 541, 569, 571, 587, 599, 601, 613, 617, 619,
631, 641, 643, 661, 673, 691, 709, 719, 739, 743, 751, 757, 787, 809, 811, 821,
823, 827, 829, 859, 863, 877, 883, 887, 907, 911, 919, 937, 947, 953, 967, 971,
991, 1009, 1019, 1021, 1031, 1039, 1051, 1061, 1069, 1109, 1123, 1151, 1171, 
1181, 1187, 1193, 1201, 1217]


 Among those that do, the largest period is the square-root of, 1033,

    whose period is, 50

                        Here are all the successful ones

                        2, : , [[1, 9], [6, 44, 6, 10]]

                            3, : , [[1, 5], [8, 6]]

                             5, : , [[2], [16, 4]]

                          7, : , [[2], [6, 20, 6, 4]]

                             11, : , [[3], [12, 6]]

                               13, : , [[3], [6]]

                             17, : , [[4], [32, 8]]

                                  19, : , FAIL

                         23, : , [[4], [5, 152, 5, 8]]

                              29, : , [[5], [10]]

                         31, : , [[5], [7, 88, 7, 10]]

                            37, : , [[6], [48, 12]]

                      41, : , [[6], [9, 4, 11, 4, 9, 12]]

                                  43, : , FAIL

                          47, : , [[6], [4, 5, 4, 12]]

                              53, : , [[7], [14]]

                       59, : , [[7], [5, 4, 6, 4, 5, 14]]

                           61, : , [[7], [4, 4, 14]]

                                  67, : , FAIL

                                  71, : , FAIL

                                  73, : , FAIL

                          79, : , [[8], [4, 7, 4, 16]]

                            83, : , [[9], [36, 18]]

                             89, : , [[9], [9, 18]]

                                  97, : , FAIL

                           101, : , [[10], [80, 20]]

                  103, : , [[10], [26, 4, 6, 9, 6, 4, 26, 20]]

                107, : , [[10], [11, 6, 10, 82, 10, 6, 11, 20]]

                                 109, : , FAIL

                                 113, : , FAIL

                                 127, : , FAIL

                     131, : , [[11], [8, 4, 44, 4, 8, 22]]

                  137, : , [[11], [5, 5, 4, 10, 4, 5, 5, 22]]

                                 139, : , FAIL

                                 149, : , FAIL

                                 151, : , FAIL

                                 157, : , FAIL

                                 163, : , FAIL

                        167, : , [[12], [4, 11, 4, 24]]

                             173, : , [[13], [26]]

                                 179, : , FAIL

                       181, : , [[13], [8, 4, 4, 8, 26]]

                                 191, : , FAIL

                     193, : , [[13], [4, 8, 13, 8, 4, 26]]

                           197, : , [[14], [112, 28]]

          199, : , [[14], [37, 8, 9, 5, 18, 56, 18, 5, 9, 8, 37, 28]]

                                 211, : , FAIL

                        223, : , [[14], [4, 13, 4, 28]]

                           227, : , [[15], [60, 30]]

                             229, : , [[15], [30]]

                           233, : , [[15], [15, 30]]

                                 239, : , FAIL

                                 241, : , FAIL

                                 251, : , FAIL

                           257, : , [[16], [128, 32]]

    263, : , [[16], [18, 9, 5, 14, 4, 6, 5, 31, 5, 6, 4, 14, 5, 9, 18, 32]]

                     269, : , [[16], [9, 4, 31, 4, 9, 32]]

                                 271, : , FAIL

                                 277, : , FAIL

      281, : , [[16], [5, 16, 7, 6, 26, 19, 536, 19, 26, 6, 7, 16, 5, 32]]

       283, : , [[16], [4, 4, 6, 14, 10, 4, 15, 4, 10, 14, 6, 4, 4, 32]]

                             293, : , [[17], [34]]

                                 307, : , FAIL

           311, : , [[17], [6, 13, 8, 4, 4, 16, 4, 4, 8, 13, 6, 34]]

                          313, : , [[17], [5, 5, 34]]

                          317, : , [[17], [4, 4, 34]]

331, : , [[18], [20, 5, 4, 6, 9, 33, 4, 4, 4, 5, 5, 6, 17, 6, 5, 5, 4, 4, 4, 33,

    9, 6, 4, 5, 20, 36]]

                                 337, : , FAIL

              347, : , [[18], [6, 10, 4, 4, 17, 4, 4, 10, 6, 36]]

                                 349, : , FAIL

       353, : , [[18], [5, 53, 4, 6, 4, 13, 37, 13, 4, 6, 4, 53, 5, 36]]

                        359, : , [[18], [4, 17, 4, 36]]

               367, : , [[19], [25, 9, 31, 4904, 31, 9, 25, 38]]

             373, : , [[19], [12, 5, 21, 12, 9, 12, 21, 5, 12, 38]]

                                 379, : , FAIL

                        383, : , [[19], [7, 312, 7, 38]]

                                 389, : , FAIL

                      397, : , [[19], [4, 12, 12, 4, 38]]

                           401, : , [[20], [160, 40]]

                                 409, : , FAIL

                                 419, : , FAIL

                                 421, : , FAIL

                                 431, : , FAIL

                                 433, : , FAIL

                        439, : , [[20], [4, 19, 4, 40]]

                           443, : , [[21], [84, 42]]

                           449, : , [[21], [21, 42]]

       457, : , [[21], [10, 6, 5, 8, 13, 5, 84, 5, 13, 8, 5, 6, 10, 42]]

                          461, : , [[21], [8, 8, 42]]

463, : ,

    [[21], [7, 5, 8, 9, 4, 37, 11, 8, 7, 85, 7, 8, 11, 37, 4, 9, 8, 5, 7, 42]]

                                 467, : , FAIL

                                 479, : , FAIL

                                 487, : , FAIL

                                 491, : , FAIL

           499, : , [[22], [11, 4, 4, 5, 4, 177, 4, 5, 4, 4, 11, 44]]

                                 503, : , FAIL

                                 509, : , FAIL

                                 521, : , FAIL

                                 523, : , FAIL

                                 541, : , FAIL

547, : ,

    [[23], [10, 12, 4, 30, 15, 8, 6, 20, 23, 20, 6, 8, 15, 30, 4, 12, 10, 46]]

                          557, : , [[23], [6, 6, 46]]

                     563, : , [[23], [5, 8, 94, 8, 5, 46]]

                                 569, : , FAIL

                                 571, : , FAIL

                           577, : , [[24], [192, 48]]

                                 587, : , FAIL

 593, : , [[24], [11, 10, 6, 7, 4, 5, 9, 4, 193, 4, 9, 5, 4, 7, 6, 10, 11, 48]]

                                 599, : , FAIL

                                 601, : , FAIL

   607, : , [[24], [6, 14, 7, 4, 15, 7, 12, 24, 12, 7, 15, 4, 7, 14, 6, 48]]

                                 613, : , FAIL

                                 617, : , FAIL

                                 619, : , FAIL

                                 631, : , FAIL

                                 641, : , FAIL

                                 643, : , FAIL

                 647, : , [[25], [9, 23, 8, 406, 8, 23, 9, 50]]

                    653, : , [[25], [7, 18, 204, 18, 7, 50]]

                     659, : , [[25], [5, 4, 24, 4, 5, 50]]

                                 661, : , FAIL

                                 673, : , FAIL

                           677, : , [[26], [208, 52]]

 683, : , [[26], [29, 5, 70, 10, 35, 4, 26, 836, 26, 4, 35, 10, 70, 5, 29, 52]]

                                 691, : , FAIL

             701, : , [[26], [8, 10, 41, 4, 51, 4, 41, 10, 8, 52]]

                                 709, : , FAIL

                                 719, : , FAIL

                        727, : , [[26], [4, 25, 4, 52]]

                             733, : , [[27], [54]]

                                 739, : , FAIL

                                 743, : , FAIL

                                 751, : , FAIL

                                 757, : , FAIL

                       761, : , [[27], [6, 4, 4, 6, 54]]

                       769, : , [[27], [5, 8, 8, 5, 54]]

                          773, : , [[27], [4, 4, 54]]

                                 787, : , FAIL

797, : ,

    [[28], [17, 13, 16, 81, 4, 7, 8, 5, 55, 5, 8, 7, 4, 81, 16, 13, 17, 56]]

                                 809, : , FAIL

                                 811, : , FAIL

                                 821, : , FAIL

                                 823, : , FAIL

                                 827, : , FAIL

                                 829, : , FAIL

                        839, : , [[28], [4, 27, 4, 56]]

853, : , [[29],

    [19, 9, 4, 18, 7, 120, 103, 4, 4, 13, 4, 4, 103, 120, 7, 18, 4, 9, 19, 58]]

                    857, : , [[29], [14, 7, 116, 7, 14, 58]]

                                 859, : , FAIL

                                 863, : , FAIL

                                 877, : , FAIL

                    881, : , [[29], [5, 4, 6, 6, 4, 5, 58]]

                                 883, : , FAIL

                                 887, : , FAIL

                                 907, : , FAIL

                                 911, : , FAIL

                                 919, : , FAIL

                   929, : , [[30], [8, 11, 5, 5, 11, 8, 60]]

                                 937, : , FAIL

941, : , [[30], [5, 4, 11, 9, 20, 5, 156, 13, 4, 4, 980, 4, 4, 13, 156, 5, 20,

    9, 11, 4, 5, 60]]

                                 947, : , FAIL

                                 953, : , FAIL

                                 967, : , FAIL

                                 971, : , FAIL

      977, : , [[31], [15, 7, 44, 4, 10, 6, 30, 6, 10, 4, 44, 7, 15, 62]]

                 983, : , [[31], [11, 11, 4, 6, 4, 11, 11, 62]]

                                 991, : , FAIL

                   997, : , [[31], [6, 4, 20, 20, 4, 6, 62]]

                                 1009, : , FAIL

                       1013, : , [[31], [4, 4, 4, 4, 62]]

                                 1019, : , FAIL

                                 1021, : , FAIL

                                 1031, : , FAIL

1033, : , [[32], [28, 7, 4, 8, 13, 48, 5, 4, 8, 5, 4, 4, 7, 4, 4, 8, 7, 4, 5,

    14, 8, 4, 146, 114, 16, 114, 146, 4, 8, 14, 5, 4, 7, 8, 4, 4, 7, 4, 4, 5, 8,

    4, 5, 48, 13, 8, 4, 7, 28, 64]]

                                 1039, : , FAIL

         1049, : , [[32], [10, 13, 16, 5, 5, 31, 5, 5, 16, 13, 10, 64]]

                                 1051, : , FAIL

                                 1061, : , FAIL

1063, : , [[32],

    [6, 6, 10, 14, 4, 28, 86, 4, 4, 31, 4, 4, 86, 28, 4, 14, 10, 6, 6, 64]]

                                 1069, : , FAIL

                        1087, : , [[32], [4, 31, 4, 64]]

                          1091, : , [[33], [132, 66]]

                             1093, : , [[33], [66]]

                           1097, : , [[33], [33, 66]]

 1103, : , [[33], [18, 4, 11, 30, 4, 11, 4, 131, 4, 11, 4, 30, 11, 4, 18, 66]]

                                 1109, : , FAIL

                1117, : , [[33], [9, 8, 21, 4, 4, 21, 8, 9, 66]]

                                 1123, : , FAIL

                          1129, : , [[33], [6, 6, 66]]

                                 1151, : , FAIL

1153, : , [[33], [4, 21, 6, 29, 7, 8, 6, 10, 4, 10, 27, 8, 46, 5, 12, 7, 4, 11,

    12, 26, 22, 8, 16, 8, 22, 26, 12, 11, 4, 7, 12, 5, 46, 8, 27, 10, 4, 10, 6,

    8, 7, 29, 6, 21, 4, 66]]

1163, : , [[34], [38, 4, 10, 4, 7, 4, 7, 6, 14, 5, 13, 6, 6, 67, 6, 6, 13, 5,

    14, 6, 7, 4, 7, 4, 10, 4, 38, 68]]

                                 1171, : , FAIL

                                 1181, : , FAIL

                                 1187, : , FAIL

                                 1193, : , FAIL

                                 1201, : , FAIL

1213, : , [[34],

    [4, 4, 4, 4, 7, 6, 8, 158, 5, 12, 278, 12, 5, 158, 8, 6, 7, 4, 4, 4, 4, 68]

    ]

                                 1217, : , FAIL

                        1223, : , [[34], [4, 33, 4, 68]]

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

[4, 2, 2, 4, 2, 1, 2, FAIL, 4, 1, 4, 2, 6, FAIL, 4, 1, 6, 3, FAIL, FAIL, FAIL,
4, 2, 2, FAIL, 2, 8, 8, FAIL, FAIL, FAIL, 6, 8, FAIL, FAIL, FAIL, FAIL, FAIL, 4
, 1, FAIL, 5, FAIL, 6, 2, 12, FAIL, 4, 2, 1, 2, FAIL, FAIL, FAIL, 2, 16, 6, 
FAIL, FAIL, 14, 14, 1, FAIL, 12, 3, 3, 26, FAIL, 10, FAIL, 14, 4, 8, 10, FAIL,
4, FAIL, 5, 2, FAIL, FAIL, FAIL, FAIL, FAIL, 4, 2, 2, 14, 3, 20, FAIL, FAIL, 
FAIL, FAIL, 12, FAIL, FAIL, FAIL, FAIL, FAIL, 18, 3, 6, FAIL, FAIL, 2, FAIL, 18
, FAIL, FAIL, 16, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, 8, 6, 6, FAIL, FAIL, 2, 
16, FAIL, 10, FAIL, FAIL, 4, 1, FAIL, FAIL, FAIL, FAIL, 5, 5, 3, FAIL, 18, FAIL
, FAIL, FAIL, FAIL, FAIL, FAIL, 4, 20, 6, FAIL, FAIL, FAIL, 7, FAIL, FAIL, FAIL
, FAIL, FAIL, 7, FAIL, 22, FAIL, FAIL, FAIL, FAIL, 14, 8, FAIL, 7, FAIL, 5, 
FAIL, FAIL, FAIL, 50, FAIL, 12, FAIL, FAIL, 20, FAIL, 4, 2, 1, 2, 16, FAIL, 9,
FAIL, 3, FAIL, 46, 28, FAIL, FAIL, FAIL, FAIL, FAIL, 22, FAIL, 4]


       This ends this article, that took, 351.394, seconds to generate.