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For the corresponding y values see A336793.

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

The corresponding y values are given in A339882.

The Diophantine equation x^2 - p*y^2 = -2, of discriminant Disc = 4*p > 0 (indefinite binary quadratic form), with prime p can have proper solutions (gcd(x, y) = 1) only for primes p = 2 and p == 3 (mod 8) by parity arguments.

There are no improper solutions (with g >= 2, g^2 does not divide 2).

The prime p = 2 has just one infinite family of proper solutions with nonnegative x values. The fundamental proper solutions for p = 2 is (0, 1).

If a prime p congruent to 3 modulo 8, (p(n) = A007520(n)) has a solution then it can have only one infinite family of (proper) solutions with positive x value.

This family is self-conjugate (also called ambiguous, having with each solution (x, y) also (x, -y) as solution). This follows from the fact that there is only one representative parallel primitive form (rpapf), namely F_{pa(n)} = [-2, 2, -(p(n) - 1)/2].

The reduced principal form of Disc(n) = 4*p(n) is F_{p(n)} = [1, 2*s(n), -(p(n) - s(n)^2)], with s(n) = A000194(n'(n)), if p(n) = A000037(n'(n)). The corresponding (reduced) principal cycle has length L(n) = 2*A307372(n'(n)).

The number of all cycles, the class number, for Disc(n) is h(n'(n)) = A324252(n'(n)), Note that in the Buell reference, Table 2B in Appendix 2, p. 241, all Disc(n) <= 4*1051 = 4204 have class number 2, except for p = 443, 499, 659 (Disc = 1772, 1996, 26).

See the W. Lang link Table 1 for some principal reduced forms F_{p(n)} (there for p(n) in the D-column, and F_p is called FR(n)) with their t-tuples, giving the automporphic matrix Auto(n) = R(t_1) R(t_1) ... R(t_{L(n)}), where R(t) := Matrix([[0, -1], [1, t]]), and the length of the principal cycle L(n) given above, and in Table 2 for CR(n).

To prove the existence of a solution one would have to show that the rpapf F{pa(n)} is properly equivalent to the principal form F_{pa(n)}.