A125612 a(n) is the smallest prime p such that 11^n divides p^10 - 1.

2, 3, 2663, 45989, 275393, 2120879, 28723679, 174625993, 4715895383, 24262286441, 1194631280321, 3143820659087, 138090848575723, 488581592070877, 6266190914259137, 367597838908577287, 10866698414795559631, 19697814061539637951, 19697814061539637951, 3824465353837845574717, 14852046860008834240157

Offset

1,1

Comments

a(n) is the smallest 10th root of unity (mod 11^n) that is prime. - Robert Israel, Jan 14 2024

Links

Robert Israel, Table of n, a(n) for n = 1..954

W. Keller and J. Richstein, Fermat quotients that are divisible by p.

Maple

f:= proc(n) local R, r, i;
  R:= sort(map(rhs@op, [msolve(x^10=1, 11^n)]));
  for i from 0 do
    for r in R do
      if isprime(11^n * i + r) then return 11^n * i + r fi
  od od;
end proc:
map(f, [$1..20]); # Robert Israel, Jan 14 2024

Mathematica

spp[n_]:=Module[{p=2, c=11^n}, While[PowerMod[p, 10, c]!=1, p=NextPrime[p]]; p]; Array[spp, 16] (* Harvey P. Dale, Aug 08 2019 *)

Prog

(PARI) \\ See A125609
(Python)
from itertools import count
from sympy import nthroot_mod, isprime
def A125612(n):
    m = 11**n
    r = sorted(nthroot_mod(1, 10, m, all_roots=True))
    for i in count(0, m):
        for p in r:
            if isprime(i+p): return i+p # Chai Wah Wu, May 02 2024

Crossrefs

Cf. A125609 = Smallest prime p such that 3^n divides p^2 - 1. Cf. A125610 = Smallest prime p such that 5^n divides p^4 - 1. Cf. A125611 = Smallest prime p such that 7^n divides p^6 - 1. Cf. A125632 = Smallest prime p such that 13^n divides p^12 - 1. Cf. A125633 = Smallest prime p such that 17^n divides p^16 - 1. Cf. A125634 = Smallest prime p such that 19^n divides p^18 - 1. Cf. A125635 = Smallest prime p such that 257^n divides p^256 - 1.

Sequence in context: A137321 A066848 A324310 * A185156 A235935 A182383

Adjacent sequences: A125609 A125610 A125611 * A125613 A125614 A125615

Keyword

nonn

Author

Alexander Adamchuk, Nov 28 2006

Extensions

More terms from Ryan Propper, Jan 03 2007

More terms from Martin Fuller, Jan 11 2007

More terms from Robert Israel, Jan 14 2024

Status

approved