A339639 a(n) is the sum of the Wieferich and Wall-Sun-Sun residues of prime(n).

3, 6, 20, 35, 110, 78, 493, 114, 736, 783, 961, 518, 2542, 2752, 2820, 3392, 1062, 5124, 1139, 4047, 8322, 5372, 5727, 979, 9118, 19089, 8343, 3959, 10137, 16159, 3937, 10611, 15207, 20433, 32184, 17516, 19782, 37001, 15197, 23009, 40096, 50499, 27504, 26055

Offset: 1

Author

Felix Fröhlich, Dec 11 2020

Keywords

• nonn

Comments

If a(n) = 0 then prime(n) is both a Wieferich prime (A001220) and a Wall-Sun-Sun (Fibonacci-Wieferich) prime.

If the first case of Fermat's last theorem fails for a prime p, that prime is both a Wieferich prime (cf. Wieferich, 1909) and a Wall-Sun-Sun prime (cf. Sun, 1992).

Links

• Zhi-Wei Sun, Fibonacci numbers and Fermat's last theorem, Acta Arithemtica, Vol. 60, No. 4 (1992), 371-388.
• A. Wieferich, Zum letzten Fermat'schen Theorem, Journal für die reine und angewandte Mathematik, 136 (1909), 293-302, DOI:10.1515/crll.1909.136.293.

Crossrefs

Cf. A001220, A113650, A196202.

Programs

PARI

a(n) = my(p=prime(n)); lift(Mod([1, 1; 1, 0]^(p-kronecker(p, 5)), p^2)[1, 2]) + lift(Mod(2, p^2)^(p-1)) - 1

Formula

a(n) = A196202(n) + A113650(n) - 1.