A125611 a(n) is the smallest prime p such that 7^n divides p^6 - 1.

2, 19, 19, 3449, 32261, 152617, 3294173, 3376853, 135967277, 135967277, 7909306973, 92233439147, 115385868869, 1356446145697, 56020344873707, 56020344873707, 930522055948829, 9116268492336169, 10744682090246617

Offset: 1

Author

Alexander Adamchuk, Nov 28 2006

Keywords

• nonn

Comments

a(n) is the smallest 6th root of unity (mod 7^n) that is prime. - Robert Israel, Jan 14 2024

Links

• Robert Israel, Table of n, a(n) for n = 1..1000
• Wilfrid Keller and Jörg Richstein, Fermat quotients that are divisible by p, 2014. [Wayback Machine link]

Crossrefs

Cf. A125609, A125610, A125612, A125632, A125633, A125634, A125635.

Programs

Maple

f:= proc(n) local R, r, i;
  R:= sort(map(rhs@op, [msolve(x^6=1, 7^n)]));
  for i from 0 do
    for r in R do
      if isprime(7^n * i + r) then return 7^n * i + r fi
  od od;
end proc:
map(f, [$1..30]); # Robert Israel, Jan 14 2024

PARI

\\ See A125609

Python

from itertools import count
from sympy import nthroot_mod, isprime
def A125611(n):
    m = 7**n
    r = sorted(nthroot_mod(1,6,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