A328138 Numbers m that divide 9^m + 8.

1, 17, 803, 1241, 20264753, 28214180783393, 228454543831049

Offset: 1

Author

Juri-Stepan Gerasimov, Oct 04 2019

Keywords

• nonn
• more

Comments

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

No term can be a multiple of 2, 3, 5, 7, or 13. Also 4879573990210017348077958628152400091281634488825721395187 is a term. - Giovanni Resta, Oct 07 2019

Also 6788776064693081883870036833 is a term. - Max Alekseyev, Dec 27 2024

Crossrefs

Subsequence of A008364.

Solutions to k^m == k-1 (mod m): 1 (k = 1), A006521 (k = 2), A015973 (k = 3), A327840 (k = 4), A123047 (k = 5), A327943 (k = 6), A328033 (k = 7), A327468 (k = 8), this sequence (k = 9).

Cf. A253212 (9^n + 8).

Programs

Magma

[1] cat [n: n in [1..10^8] | Modexp(9, n, n) + 8 eq n];

PARI

isok(n) = Mod(9, n)^n==-8; \\ Michel Marcus, Oct 05 2019

Formula

a(n) > 15n for large enough n. (Surely the sequence grows superlinearly, but I can't prove it.) - Charles R Greathouse IV, Dec 27 2024

Extensions

a(7) from Giovanni Resta confirmed and a(6) added by Max Alekseyev, Dec 27 2024