A336789 - OEIS

A336789

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

%I #14 Feb 27 2021 21:49:31

%S 1,7,47,193,3383,9041,20687,731153,8808724183,98546821297,

%T 2208304390649,19569442212887,162848901149273,311991807873328639,

%U 1023490545293318137,1419456983764900351,13170848364266136042527,1276022762028643136592313,14225223924067129319855681

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

%e 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.

%e 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.

%e From _Jon E. Schoenfield_, Feb 24 2021: (Start)

%e 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:

%e .

%e x values satisfying minimal

%e D x^2 - D*y^2 = 2 y value record

%e --- ------------------------ ------- ------

%e 2 1, 7, 41, 239, 1393, ... 1 *

%e 7 1, 17, 271, 4319, ... 1

%e 23 1, 49, 2351, 112799, ... 1

%e 31 7, 21287, 64712473, ... 7 *

%e 47 1, 97, 9311, 893759, ... 1

%e 71 7, 48727, 339139913, ... 7

%e 79 1, 161, 25759, ... 1

%e 103 47, 21387679, ... 47 *

%e ...

%e 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)

%Y Cf. A033315, A336788.

%K nonn

%O 1,2

%A _Christine Patterson_, Oct 14 2020

%E Example section edited by _Jon E. Schoenfield_, Feb 24 2021