A125646 Smallest odd prime base q such that p^5 divides q^(p-1) - 1, where p = prime(n).

97, 487, 14557, 32261, 275393, 220861, 15541, 2342959, 1051847, 24639193, 40373093, 70697317, 31851901, 47289133, 456330179, 10000453, 154075723, 130702609, 304154189, 143584109, 183298237, 79451167, 1058782027, 352845203, 567620413, 4592184511, 5890772963, 9651540247, 4081988041, 4772484029

Offset: 1

Author

Alexander Adamchuk, Nov 29 2006

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..500
• W. Keller and J. Richstein, Fermat quotients that are divisible by p.

Crossrefs

Cf. A125609, A125610, A125611, A125612, A125632, A125633, A125634, A125635, A125636, A125637, A125645, A125647, A125648, A125649.

Programs

Maple

f:= proc(n) local p,k,j,q,R;
  p:= ithprime(n);
  R:= sort(map(rhs@op, [msolve(q^(p-1)-1, p^5)]));
  for k from 0 do
    for j in R do
      q:= k*p^5+j;
      if isprime(q) then return q fi;
    od
 od
end proc:
map(f, [$1..100]); # Robert Israel, Apr 11 2019

Mathematica

Do[p = Prime[n]; q = 2; While[PowerMod[q, p-1, p^5] != 1, q = NextPrime[q]]; Print[q], {n, 100}] (* Ryan Propper, Mar 31 2007 *)

PARI

{ a(n) = local(p,x,y); if(n==1,return(97)); p=prime(n); x=znprimroot(p^5)^(p^4); vecsort( vector(p-1,i, y=lift(x^i);while(!isprime(y),y+=p^5);y ) )[1] } \\ Max Alekseyev, May 30 2007

Python

from itertools import count
from sympy import nthroot_mod, isprime, prime
def A125646(n):
    m = (p:=prime(n))**5
    r = sorted(nthroot_mod(1,p-1,m,all_roots=True))
    for i in count(0,m):
        for a in r:
            if isprime(i+a): return i+a # Chai Wah Wu, May 02 2024

Extensions

More terms from Ryan Propper, Mar 31 2007

More terms from Max Alekseyev, May 30 2007