A345753 Euler-Fibonacci pseudoprimes: odd composites k such that F(k) == 5^((k-1)/2) == +-1 (mod k), where F(k) = A000045(k), the Fibonacci numbers.

146611, 252601, 399001, 512461, 556421, 852841, 1024651, 1193221, 1314631, 1857241, 1909001, 2100901, 2165801, 2603381, 2704801, 3470921, 3828001, 3942271, 4504501, 5049001, 5148001, 5481451, 6189121, 6840001, 7267051, 7519441, 7879681, 8086231, 8341201, 8719921, 9439201, 9863461

Offset: 1

Author

Amiram Eldar and Thomas Ordowski, Jun 26 2021

Keywords

• nonn

Comments

If p is an odd prime except 5, then F(p) == 5^((p-1)/2) == +-1 (mod p).

All terms found satisfy the congruence F(k) == 5^((k-1)/2) == 1 (mod k). They are a proper subset of A094394.

Are there odd composites m such that F(m) == 5^((m-1)/2) == -1 (mod m)? They are a proper subset (maybe empty) of A094395 (they are not in the database, below 4*10^9).

Crossrefs

Cf. A000045, A094394, A094395, A262052.

Programs

Mathematica

Select[Range[1, 10^6, 2], CompositeQ[#] && MemberQ[{1, # - 1}, PowerMod[5, (# - 1)/2, #]] && Divisible[5^((# - 1)/2) - Fibonacci[#], #] &]