A244801 Smallest m such that for the prime p = prime(n) the congruence F_(p-(p/5)) == mp (mod p^2) holds (i.e., smallest m such that prime(n) is a near-Wall-Sun-Sun prime), where F_k is the k-th Fibonacci number and (p/5) is the Legendre symbol.

1, 1, 1, 3, 5, 3, 16, 3, 15, 26, 25, 13, 39, 39, 16, 28, 10, 48, 7, 55, 58, 49, 21, 5, 37, 97, 22, 24, 26, 60, 13, 64, 58, 117, 120, 60, 44, 160, 44, 130, 174, 131, 94, 31, 141, 5, 112, 3, 154, 18, 29, 5, 182, 250, 2, 105

Offset: 1

Author

Felix Fröhlich, Jul 06 2014

Keywords

• nonn

Comments

A value of 0 indicates a Wall-Sun-Sun prime. No such prime is known and if one exists it is > 4*10^16 (cf. PrimeGrid WSS statistics).

Links

• Felix Fröhlich, Table of n, a(n) for n = 1..291043
• F. G. Dorais and D. Klyve, A Wieferich Prime Search up to 6.7 x 10^15, J. Integer Seq. Volume 14, Issue 9 (2011).
• R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp. 76 (2007), 2087-2094.
• PrimeGrid, Wall-Sun-Sun Prime Search statistics
• PrimeGrid, Welcome to the Wall-Sun-Sun prime search

Crossrefs

Cf. A000045, A113650, A195988.

Programs

Mathematica

A= 0; p = 0; While[A < 200, p = NextPrime[p];  A= Mod[(Fibonacci[p-JacobiSymbol[p,5]])/p, p]; Print[A]] (* Javier Rivera Romeu, Jan 11 2022 *)

PARI

forprime(p=2, 10^2, a=fibonacci(p-kronecker(p, 5))%p^2; a=a/p; print1(a, ", "))

Sage

A, p = 0, 0
while A <200:
  p = next_prime(p)
  A = (fibonacci(p-legendre_symbol(p, 5))/p)%p
  print(A, end=", ") #Javier Rivera Romeu, Jan 08 2022