Conjectures of Golomb and Dasgupta

Thomas R. Nicely

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Copyright © 2005 Thomas R. Nicely. All rights reserved. This document may be reproduced and distributed for educational and non-profit purposes. No warranties expressed or implied.

Last updated 0409 EDT 6 September 2005. Originally posted 6 September 2005.


Anirban Dasgupta (Purdue University) attributes to Solomon W. Golomb the conjecture that for every integer k > 1, n/pi(n) = k for some integer n > 1. Here pi(n) is the prime counting function, the count of prime numbers <= n. No proof of this conjecture is known.

By analogy, Dasgupta conjectures that for every integer k > 2, n/pi_2(n) = k for some integer n > 2. Here pi_2(n) is the twin-prime counting function, the count of twin-prime pairs (q, q+2) such that q is prime, q+2 is prime, and q <= n. No proof of this conjecture is known.

This document presents some computational evidence, generated by the author, in support of these two conjectures.

In the first table, the third and fourth columns display the smallest and largest values obtained for n, such that n/pi(n) = k, where k is given in the first column. The second column gives the total number m of distinct values of n for which n/pi(n) = k.

=======================================================
                 GOLOMB'S CONJECTURE
=======================================================
     k     m                n_min                n_max
=======================================================
     2     4                    2                    8
     3     3                   27                   33
     4     3                   96                  120
     5     6                  330                  360
     6     7                 1008                 1134
     7     6                 3059                 3094
     8     6                 8408                 8472
     9     3                23526                24300
    10     9                64540                64720
=======================================================
In the second table, the third and fourth columns display the smallest and largest values obtained for n, such that n/pi_2(n) = k, where k is given in the first column. The second column gives the total number m of distinct values of n for which n/pi_2(n) = k.
=======================================================
                DASGUPTA'S CONJECTURE
=======================================================
     k     m                n_min                n_max
=======================================================
     3     2                    3                    6
     4     3                    4                   12
     5     3                   10                   20
     6     2                   24                   30
     7     3                   28                   42
     8     2                   40                   48
     9     3                   54                   72
    10     2                   70                   80
    11     2                   88                  110
    12     2                   96                  120
    13     3                  130                  156
    14     4                  168                  210
    15     4                  225                  285
    16     2                  304                  320
    17     2                  340                  357
    18     1                  378                  378
    19     2                  399                  437
    20     2                  460                  480
    21     2                  504                  525
    22     3                  550                  660
    23     2                  598                  690
    24     1                  720                  720
    25     1                  750                  750
    26     2                  780                  910
    27     1                  945                  945
    28     3                  980                 1120
    29     4                 1015                 1334
    30     3                 1260                 1500
    31     2                 1426                 1550
    32     6                 1600                 2144
    33    11                 1848                 2343
    34     1                 2448                 2448
    35     3                 2520                 2730
    36     1                 2880                 2880
    37     4                 2960                 3589
    38    11                 3116                 4370
    39     4                 4719                 4836
    40     1                 5160                 5160
    41     5                 5371                 5699
    42     1                 6006                 6006
    43     6                 6407                 6966
    44     2                 7172                 7216
    45     2                 7740                 7785
    46     2                 8142                 8234
    47     1                 8695                 8695
    48     5                 9216                 9552
    49     1                10633                10633
    50     2                11050                11300
    51     8                11679                12291
    52     1                12688                12688
    53     1                13144                13144
    54     6                13554                14634
    55     1                14960                14960
    56     4                15512                15792
    57     5                16587                18411
    58     2                18850                18966
    59    16                20355                23069
    60     1                24240                24240
    61     1                24766                24766
    62     4                25792                26040
    63     5                26649                28854
    64     2                29824                29888
    65    16                31070                35165
    66     1                36630                36630
    67     9                38324                39396
    68     2                40392                40460
    69     2                42987                43056
    70     1                45500                45500
    71    12                47286                50339
    72     5                52848                53640
    73     8                55188                56940
    74     4                60014                60310
    75     5                61950                62325
    76     4                64448                64676
=======================================================
These results will be extended as time and available computational power allow. Note that the format of this document and its tables is subject to change.