A241014 Let p be the n-th prime, then a(n) = A/p where A is the smallest number (in absolute value) such that F_{p-(p/5)} == A (mod p^2) with F_n = A000045(n) and (p/5) the Legendre symbol.

1, 1, 1, 3, 5, 3, -1, 3, -8, -3, -6, 13, -2, -4, 16, -25, 10, -13, 7, -16, -15, -30, 21, 5, 37, -4, 22, 24, 26, -53, 13, 64, 58, -22, -29, 60, 44, -3, 44, -43, -5, -50, 94, 31, -56, 5, -99, 3, -73, 18, 29, 5, -59, -1, 2

Offset: 1

Author

Felix Fröhlich, Aug 13 2014

Keywords

• sign

Comments

a(n) is the smallest A such that p is a near-Wall-Sun-Sun prime. A gives the value of F_p-(p/5) modulo p^2 and a value of 0 would indicate a Wall-Sun-Sun prime. A244801 is similar but always gives the positive A, while this sequence gives A with the smallest absolute value.

a(1), with p=2, is technically ambiguous between 1 and -1, so a(1)=1 is by convention. Clearly this cannot happen for n>1 (where p^2 is odd). - Jeppe Stig Nielsen, Sep 09 2021

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp., 76 (2007), 2087-2094.

Crossrefs

Cf. A113650, A195988, A244801, A339855, A347565.

Programs

Mathematica

Array[(#3 - #2 Boole[#3 > #2/2])/#1 & @@ {#, #^2, Mod[Fibonacci[# - KroneckerSymbol[#, 5]], #^2]} &@ Prime[#] &, 55] (* Michael De Vlieger, Sep 08 2021 *)

PARI

forprime(p=2, 1e2, a=fibonacci(p-kronecker(p, 5))%p^2; if(a>p^2/2, a-=p^2); a=a/p; print1(a, ", "))

PARI

a(n)=my(p=prime(n)); centerlift(((Mod([1, 1; 1, 0], p^2))^(p-kronecker(p,5))))[1, 2]/p \\ Charles R Greathouse IV, Aug 21 2014