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%I #22 Feb 14 2021 01:38:15
%S 3,19,43,67,139,211,331,379,571,739,859,1051,1291,1531,1579,1699,2011,
%T 2731,3019,3259,3691,3931,5419,5659,5779,6211,6379,6451,8779,9619,
%U 10651,16699,17851,18379,21739,25939,32971,42331,42571,44851,50131,53299,55819,56611,60811,61051,73459,76651,90619,90931
%N Values of odd prime numbers, D, for incrementally largest values of minimal positive y satisfying the equation x^2 - D*y^2 = -2.
%C For the corresponding y values see A336793.
%C For solutions of this Diophantine equation it is sufficient to consider the odd primes p(n) := A007520(n), for n >= 1, the primes 3 (mod 8). Also prime 2 has the fundamental solution (x, y) = (0, 1). If there is a solution for p(n) then there is only one infinite family of solutions because there is only one representative parallel primitive binary quadratic form for Discriminant Disc = 4*p(n) and representation k = -2. Only proper solutions can occur. The conjecture is that each p(n) leads to solutions. For the fundamental solutions (with prime 2) see A339881 and A339882. - _Wolfdieter Lang_, Dec 22 2020
%H Christine Patterson, <a href="/A336792/a336792.txt">Sage Program</a>
%e For D=3, the least positive y for which x^2-D*y^2=-2 has a solution is 1. The next prime, D, for which x^2-D*y^2=-2 has a solution is 11, but the smallest positive y in this case is also 1, which is equal to the previous record y. So 11 is not a term.
%e The next prime, D, after 11 for which x^2-D*y^2=-2 has a solution is 19 and the least positive y for which it has a solution is y=3, which is larger than 1, so it is a new record y value. So 19 is a term of this sequence and 3 is a term of A336793.
%Y Cf. A033316 (analogous for x^2-D*y^2=1), A336790 (similar sequence for x's), A336793.
%K nonn
%O 1,1
%A _Christine Patterson_, Oct 14 2020
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