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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.
2
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
1,1
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)).
LINKS
N. Osipov et al., Residues on a circle (in Russian), dxdy.ru, 2024.
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).
EXAMPLE
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).
PROG
(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, ", ")) );
CROSSREFS
Contains Fermat primes (A019434) as a subsequence.
Sequence in context: A181032 A331649 A322737 * A201107 A210902 A188803
KEYWORD
nonn
AUTHOR
STATUS
approved