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

Author

Alexander Adamchuk, Nov 28 2006

Keywords

• nonn

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.

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.

Programs

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 *)

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

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