[seqfan] Re: The Euler-Fibonacci pseudoprimes

[Ami Eldar]

The odd composites m such  that F(m) == 5^{(m-1)/2} == +-1 (mod m) are:
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, ...

Best,
Amiram

On Thu, Jun 17, 2021 at 9:48 PM Tomasz Ordowski <tomaszordowski at gmail.com>
wrote:

> 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
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>