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

1,4

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

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 *)

Prog

(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

Crossrefs

Cf. A000045, A113650, A195988.

Sequence in context: A121278 A023587 A172003 * A002586 A258811 A066845

Adjacent sequences: A244798 A244799 A244800 * A244802 A244803 A244804

Keyword

nonn

Author

Felix Fröhlich, Jul 06 2014

Status

approved