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

1,1

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).

Links

Table of n, a(n) for n=1..32.

Mathematica

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

Crossrefs

Cf. A000045, A094394, A094395, A262052.

Sequence in context: A233915 A237297 A204527 * A110084 A224587 A004670

Adjacent sequences: A345750 A345751 A345752 * A345754 A345755 A345756

Keyword

nonn

Author

Amiram Eldar and Thomas Ordowski, Jun 26 2021

Status

approved