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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.

%I A376008 #16 Sep 15 2024 11:03:12
%S A376008 3,17,251,257,433,641,1459,3457,3889,21169,39367,54001,65537,110251,
%T A376008 114689,139969,210913,246241,274177,319489,629857,746497,974849,
%U A376008 995329,1161217,1299079,1492993,1769473,2020001,2424833,2555521,2654209,5038849,5304641,5419387,5746001,6049243,6561001
%N 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.
%C A376008 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.
%C A376008 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)).
%H A376008 Max Alekseyev, <a href="/A376008/b376008.txt">Table of n, a(n) for n = 1..100</a>
%H A376008 N. Osipov et al., <a href="https://dxdy.ru/topic158356.html">Residues on a circle</a> (in Russian), dxdy.ru, 2024.
%F A376008 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).
%e A376008 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).
%o A376008 (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,", ")) );
%Y A376008 Contains Fermat primes (A019434) as a subsequence.
%Y A376008 Cf. A001918, A002326, A046144, A376010.
%K A376008 nonn
%O A376008 1,1
%A A376008 _Nikolay Osipov_, _Dmitry Petukhov_, and _Max Alekseyev_, Sep 05 2024