A376008 Primes p such that there exists a cyclic permutation of the nonzero residues modulo p such that v^2 - 4*u*w == 0 (mod p) for any three consecutive residues u,v,w.

3, 17, 251, 257, 433, 641, 1459, 3457, 3889, 21169, 39367, 54001, 65537, 110251, 114689, 139969, 210913, 246241, 274177, 319489, 629857, 746497, 974849, 995329, 1161217, 1299079, 1492993, 1769473, 2020001, 2424833, 2555521, 2654209, 5038849, 5304641, 5419387, 5746001, 6049243, 6561001

Offset: 1

Author

Nikolay Osipov, Dmitry Petukhov, and Max Alekseyev, Sep 05 2024

Keywords

• nonn

Comments

In other words, for any three consecutive residues u,v,w, the quadratic polynomial u*x^2 + v*x + w has zero discriminant modulo p.

It is shown that all suitable permutations q for prime p = a(n) can be constructed by starting with q(1) = 1, q(2) = a primitive root modulo p, and then defining q(k) = q(k-1)^2/(4*q(k-2)) mod p for k >= 3. Hence, the number of suitable permutations (up to cyclic rotations) is given by A046144(a(n)).

Examples

			For a(2) = 17, a suitable cyclic permutation is (1, 3, 15, 6, 4, 12, 9, 7, 16, 14, 2, 11, 13, 5, 8, 10).
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..100
• N. Osipov et al., Residues on a circle (in Russian), dxdy.ru, 2024.

Crossrefs

Contains Fermat primes (A019434) as a subsequence.

Cf. A001918, A002326, A046144, A376010.

Programs

PARI

forprime(p=3,10^8, s=(p-1)/znorder(Mod(2,p)); if(factor(p-1)[,1]==factor(2*s)[,1] && !(p%4==1 && s%2==1),print1(p,", ")) );

Formula

An odd prime p is a term iff for s:=(p-1)/A002326((p-1)/2), radicals of p-1 and 2s coincide, excluding the case p==1 (mod 4) and s==1 (mod 2).