A329168 -id:A329168 - cp's OEIS frontend

A329222 Numbers m that divide 5^(m + 1) + 1.

1, 2, 6, 13, 14, 174, 854, 2694, 78126, 103973, 106694, 121974, 420209, 487374, 1299374, 2174654, 3895094, 4151454, 5842214, 5951129, 6508334, 10637054, 20117894, 24482957, 31999694, 32282053, 32620202, 32872454, 34258454, 52657397, 56114618, 57679082, 65538437, 70782774, 71899526

Offset: 1

Juri-Stepan Gerasimov, Nov 08 2019

nonn

Conjecture: For k > 1, k^(m + 1) == -1 (mod m) has an infinite number of positive solutions.

Cf. A055685.

Solutions to k^(m + 1) == -1 (mod m): A296369 (k=2), A328230 (k=3), A329168 (k=4), this sequence (k=5), A329226 (k=6).

[n + 1: n in [0..2000000] | Modexp(5, n + 2, n + 1) eq n];

Select[Range[719*10^5],PowerMod[5,#+1,#]==#-1&] (* Harvey P. Dale, Jul 03 2020 *)

isok(m) = Mod(5, m)^(m+1) == -1; \\ Jinyuan Wang, Nov 16 2019

A329226 Numbers m that divide 6^(m + 1) + 1.

Original entry on oeis.org

1, 37, 16987849, 2416266949, 5995229029, 7193673829, 11465419549, 17783484529, 72155530501, 142013229529, 174523785589, 189282539137, 294183810997, 302690164297, 354613312129, 774557575609, 933821938789, 1407294504937, 1974020768389, 2112969494569, 2878251281401

Offset: 1

Juri-Stepan Gerasimov, Nov 08 2019

nonn

Conjecture: For k > 1, k^(m + 1) == -1 (mod m) has an infinite number of positive solutions.

Terms cannot be a multiple of the following primes below 100: 2, 3, 5, 7, 11, 19, 23, 29, 31, 43, 47, 53, 59, 67, 71, 79, 83. - Giovanni Resta, Nov 09 2019

Cf. A055685.

Solutions to k^(m + 1) == -1 (mod m): A296369 (k=2), A328230 (k=3), A329168 (k=4), A329222 (k=5).

[n + 1: n in [0..20000000] | Modexp(6, n + 2, n + 1) eq n];

isok(m) = Mod(6, m)^(m+1) == -1; \\ Jinyuan Wang, Nov 16 2019

a(5)-a(21) from Giovanni Resta, Nov 09 2019

cp's OEIS Frontend

A329222 Numbers m that divide 5^(m + 1) + 1.

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