[seqfan] Re: The Euler-Fibonacci pseudoprimes

[Ami Eldar]

Odd composites n such that F(n) == 5^{(n-1)/2} == -1 (mod n): there are no
such numbers below 3*10^9.

On Sat, Jun 19, 2021 at 5:52 PM Tomasz Ordowski <tomaszordowski at gmail.com>
wrote:

> Ami, thanks for finding these numbers!
>
> All pseudoprimes m found by Amiram satisfy the congruence
> F(m) == 5^{(m-1)/2} == 1 (mod m). They are a subset of A094394.
> Are there odd composites n such that F(n) == 5^{(n-1)/2} == -1 (mod n) ?
> They are a proper subset (maybe empty) of A094395:
> https://oeis.org/A094395
>
> Have a successful furter search!
>
> Tom
>
> pt., 18 cze 2021 o 01:22 Ami Eldar <amiram.eldar at gmail.com> napisał(a):
>
> > 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
> > >
> > > --
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> > >
> >
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