

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



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
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OFFSET

1,1


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


LINKS

Table of n, a(n) for n=1..50.
Christine Patterson, Sage Program


EXAMPLE

For D=3, the least positive y for which x^2D*y^2=2 has a solution is 1. The next prime, D, for which x^2D*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^2D*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.


CROSSREFS

Cf. A033316 (analogous for x^2D*y^2=1), A336790 (similar sequence for x's), A336793.
Sequence in context: A141170 A107154 A141373 * A341087 A031393 A146672
Adjacent sequences: A336789 A336790 A336791 * A336793 A336794 A336795


KEYWORD

nonn


AUTHOR

Christine Patterson, Oct 14 2020


STATUS

approved



