A336792 Values of odd prime numbers, D, for incrementally largest values of minimal positive y satisfying the equation x^2 - D*y^2 = -2.

3, 19, 43, 67, 139, 211, 331, 379, 571, 739, 859, 1051, 1291, 1531, 1579, 1699, 2011, 2731, 3019, 3259, 3691, 3931, 5419, 5659, 5779, 6211, 6379, 6451, 8779, 9619, 10651, 16699, 17851, 18379, 21739, 25939, 32971, 42331, 42571, 44851, 50131, 53299, 55819, 56611, 60811, 61051, 73459, 76651, 90619, 90931

Offset: 1

Author

Christine Patterson, Oct 14 2020

Keywords

• nonn

Comments

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

Examples

			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.
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.
		

Links

• Christine Patterson, Sage Program

Crossrefs

Cf. A033316 (analogous for x^2-D*y^2=1), A336790 (similar sequence for x's), A336793.