[seqfan] Primes p == 3 (mod 4) with a Fibonacci primitive root.
Ami Eldar
amiram.eldar at gmail.com
Sun Jan 30 11:34:47 CET 2022
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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