A336789 Incrementally largest values of minimal y satisfying the equation x^2 - D*y^2 = 2, where D is a prime number.

1, 7, 47, 193, 3383, 9041, 20687, 731153, 8808724183, 98546821297, 2208304390649, 19569442212887, 162848901149273, 311991807873328639, 1023490545293318137, 1419456983764900351, 13170848364266136042527, 1276022762028643136592313, 14225223924067129319855681

Offset: 1

Author

Christine Patterson, Oct 14 2020

Keywords

• nonn

Examples

			For D=2, the least y for which x^2 - D*y^2 = 2 has a solution is 1. The next primes, D, for which x^2 - D*y^2 = 2 has a solution are 7 and 23, but the smallest y in each of these cases is also 1, which is equal to the previous record y. So 7 and 23 are not terms of A336788.
The next prime, D, after 23 for which x^2 - D*y^2 = 2 has a solution is 31 and the least y for which it has a solution there is y=7, which is larger than 1, so it is a new record y value. So 31 is a term of A336788, and 7 is the corresponding term here.
From _Jon E. Schoenfield_, Feb 24 2021: (Start)
Primes D for which the equation x^2 - D*y^2 = 2 has integer solutions begin 2, 7, 23, 31, 47, 71, 79, 103, ...; at those values of D, the minimal y values satisfying the equation x^2 - D*y^2 = 2 begin as follows:
.
         x values satisfying     minimal
    D      x^2 - D*y^2 = 2       y value  record
  ---  ------------------------  -------  ------
    2  1, 7, 41, 239, 1393, ...      1       *
    7  1, 17, 271, 4319, ...         1
   23  1, 49, 2351, 112799, ...      1
   31  7, 21287, 64712473, ...       7       *
   47  1, 97, 9311, 893759, ...      1
   71  7, 48727, 339139913, ...      7
   79  1, 161, 25759, ...            1
  103  47, 21387679, ...            47       *
  ...
The record high minimal values of y (marked with asterisks) are the terms of this sequence. The corresponding values of D are the terms of A336788. (End)
		

Crossrefs

Cf. A033315, A336788.

Extensions

Example section edited by Jon E. Schoenfield, Feb 24 2021