[seqfan] Re: Primes p == 3 (mod 4) with a Fibonacci primitive root.

[Neil Sloane]

I have not seen any follow-up to Ami's posting, so I am taking the liberty
of adding a version of his message to A106535.

Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com



On Sun, Jan 30, 2022 at 6:16 AM Ami Eldar <amiram.eldar at gmail.com> wrote:

> Dear SeqFans,
>
> Sequence A003147, "Primes p with a Fibonacci primitive root", was defined
> in the paper:
>      Daniel Shanks, Fibonacci primitive roots, Fibonacci Quarterly, Vol.
> 10, No. 2 (1972), pp. 163-168, 181.
>      https://www.fq.math.ca/Scanned/10-2/shanks-a.pdf
>
> A second paper on this subject
>      Daniel Shanks and Larry Taylor, An Observation of Fibonacci Primitive
> Roots, Fibonacci Quarterly, Vol. 11, No. 2 (1973), pp. 159-160.
>      https://www.fq.math.ca/Scanned/11-2/shanks.pdf
> deals with terms p == 3 (mod 4) of A003147, i.e., the intersection of
> A003147 and A002145 (or A004767).
> It states that if g is a Fibonacci primitive root of a prime p such that p
> == 3 (mod 4) then g-1 and g-2 are also primitive roots of p.
>
> It seems that this sequence already exists in the OEIS:
>      A106535 Numbers k such that the smallest x > 1 for which Fibonacci(x)
> = 0 mod k is x = k - 1.
> but its definition is different, its terms are not necessarily primes, it
> does not mention A003147, nor saying that the terms are == 3 (mod 4).
> I checked and found that the first 2000 terms of A106535 are the same as
> (A003147 INTERSECT A002145).
> Can it be proved that the two sequences are the same?
>
> Best regards,
> Amiram
>
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