A239957 -id:A239957 - cp's OEIS frontend

A331506 Least primitive root g < prime(n) of the n-th prime with g a product of two Fibonacci numbers, or 0 if such a number g does not exist.

1, 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 6, 3, 5, 2, 2, 2, 2, 13, 5, 3, 2, 3, 5, 2, 5, 2, 6, 3, 3, 2, 3, 2, 2, 6, 5, 2, 5, 2, 2, 2, 21, 5, 2, 3, 2, 3, 2, 6, 3, 13, 13, 6, 3, 5, 2, 6, 5, 3, 3, 2, 5, 34, 10, 2, 3, 10, 2, 2, 3, 13, 6, 2, 2, 5, 2, 5, 3, 21, 2, 2, 13, 5, 15, 2, 3, 13, 2, 3, 2, 13, 3, 2, 10, 5, 2, 3, 2, 2

Offset: 1

Zhi-Wei Sun, Jan 18 2020

nonn

Conjecture 1: a(n) > 0 for all n > 0. In other words, for each prime p there are two Fibonacci numbers F(k) and F(m) with F(k)*F(m) < p such that F(k)*F(m) is a primitive root modulo p.
This implies that for each odd prime p there exists a Fibonacci number F(k) < p which is a quadratic nonresidue modulo p.
It seems that Conjecture 1 can be strengthened as follows: For any prime p, there is a primitive root g < p modulo p such that g/F(2) = g or g/F(3) = g/2 or g/F(4) = g/3 is a Fibonacci number. We have verified this strong version for all primes p < 5*10^9.
We also have the following conjecture similar to Conjecture 1.
Conjecture 2. For any prime p, there are two Lucas numbers L(k) and L(m) with k >= m >= 0 and L(k)*L(m) < p such that L(k)*L(m) is a primitive root modulo p.
This has been verified for all primes p < 10^9.

			a(85) = 15 with 15 = 3*5 = F(4)*F(5) a primitive root modulo prime(85) = 439.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014.
Zhi-Wei Sun, Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers, Question 350730 at MathOverflow, Jan. 19, 2020.

Cf. A000032, A000040, A000045, A001918, A239957, A241476, A241504, A241568, A241604.

p[n_]:=p[n]=Prime[n];
Dv[n_]:=Dv[n]=Divisors[p[n]-1];
ls={};
Do[If[Fibonacci[k]Fibonacci[m]=p[n],tab=Append[tab,0];Goto[bb]];Do[If[PowerMod[ff[r],Dv[n][[i]],p[n]]==1,r=r+1;Goto[aa]],{i,1,Length[Dv[n]]-1}];tab=Append[tab,ff[r]];Label[bb],{n,1,100}];Print[tab]

A344620 Primes p such that there is no positive integer a with 2*a, a^2-1 and a^2+1 not only smaller than p but also quadratic residues modulo p.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 79, 89, 97, 101, 113, 151, 173, 281, 283, 313, 461, 739, 827

Offset: 1

Zhi-Wei Sun, May 24 2021

nonn

No more terms below 10^10. For any prime p > 11, one of 1^1+1 = 2, 2^2+1 = 5 and 3^2+1 = 10 is a quadratic residue modulo p.
Conjecture: No term is greater than 827. In other words, for any prime p > 828, there is a Pythagorean triple (2*a, a^2-1, a^2+1) with 2*a, a^2-1 and a^2+1 in the set {0 < r < p: r is a quadratic residue modulo p}.
See also A344621 for a similar conjecture.

			a(5) = 13. The prime 11 is not a term since 2*2 = 4, 2^2-1 = 3 and 2^2+1 = 5 belong to the set {0 < r < 11: r is a quadratic residue modulo 11} = {1, 3, 4, 5, 9}.

Cf. A000040, A239957, A260911, A344621.

tab={}; Do[p:=p=Prime[k]; Do[If[p>2&&JacobiSymbol[2a,p]==1&&JacobiSymbol[a^2-1,p]==1&&JacobiSymbol[a^2+1,p]==1,Goto[aa]],{a,1,Sqrt[p-2]}]; tab=Append[tab,p];Label[aa],{k,1,150}]; Print[tab]

A344621 Primes p such that there is no positive integer a with 2*a, a^2-1 and a^2+1 not only smaller than p but also quadratic nonresidues modulo p.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 31, 41, 43, 47, 67, 71, 73, 97, 101, 127, 151, 157, 167, 191, 199, 239, 257, 311, 313, 367, 409, 439, 479, 521, 587, 599, 739, 839, 887, 1031, 1063, 1151, 1319, 2351, 2999, 3119

Offset: 1

Zhi-Wei Sun, May 24 2021

nonn

No more terms below 10^10.
Conjecture: No term is greater than 3119. In other words, for any prime p > 3120, there is a Pythagorean triple (2*a,a^2-1,a^2+1) with 2*a, a^2-1 and a^2+1 in the set {0 < r < p: r is a quadratic nonresidue modulo p}.
See also A344620 for a similar conjecture.

			a(5) = 13. The prime 11 is not a term since 2*3 = 6, 3^2-1 = 8 and 3^2+1 = 10 belong to the set {0 < r < 11: r is a quadratic nonresidue modulo 11} = {2, 6, 7, 8, 10}.

Cf. A000040, A239957, A260911, A344620.

tab={};Do[p:=p=Prime[k];Do[If[JacobiSymbol[2a,p]==-1&&JacobiSymbol[a^2-1,p]==-1&&JacobiSymbol[a^2+1,p]==-1,Goto[aa]],{a,1,Sqrt[p-2]}];tab=Append[tab,p];Label[aa],{k,1,450}];Print[tab]

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A331506 Least primitive root g < prime(n) of the n-th prime with g a product of two Fibonacci numbers, or 0 if such a number g does not exist.

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A344620 Primes p such that there is no positive integer a with 2*a, a^2-1 and a^2+1 not only smaller than p but also quadratic residues modulo p.

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A344621 Primes p such that there is no positive integer a with 2*a, a^2-1 and a^2+1 not only smaller than p but also quadratic nonresidues modulo p.

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