# [seqfan] The Euler-Fibonacci pseudoprimes

Tomasz Ordowski tomaszordowski at gmail.com
Thu Jun 17 07:32:39 CEST 2021

```Dear readers!

Let F(n) = Fibonacci(n) = A000045(n).

As is known, if p <> 5 is prime, then p | F(p-1) or p | F(p+1).
By Cassini's identity, if p <> 5 is prime, then p | F(p)^2+(-1)^p.
There are composite numbers n | F(n)^2+(-1)^n, namely
231, 323, 377, 442, 1378, 1443, 1551, 1891, 2737, 2834, 2849, ...
Such odd numbers are A337231.
If p <> 5 is an odd prime, then F(p) == +-1 (mod p).
Such odd pseudoprimes are A094394 and A094395.

If p <> 5 is an odd prime, then F(p) == 5^{(p-1)/2} == +-1 (mod p).
The weak Euler-Fibonacci pseudoprimes can be defined as
odd composites k such that F(k) == 5^{(k-1)/2} (mod k),
but maybe someone has already done it on the OEIS pages.
25, 75, 125, 425, 555, 625, 1625, 1875, 1891, 3125, 4375, 13161, ...
Such pseudoprimes indivisible by 5 are 1891, 13161, 13981, 68101, ...
However, this subset is also not in the OEIS. Data from Amiram Eldar.
Consider the strong Euler-Fibonacci pseudoprimes with the full condition:
F(m) == 5^{(m-1)/2} == +-1 (mod m). Are there such odd composites m?
Maybe someone will find such pseudoprimes, if they exist (in the OEIS).

Best regards,

Thomas Ordowski

```