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

%I A336789 #14 Feb 27 2021 21:49:31
%S A336789 1,7,47,193,3383,9041,20687,731153,8808724183,98546821297,
%T A336789 2208304390649,19569442212887,162848901149273,311991807873328639,
%U A336789 1023490545293318137,1419456983764900351,13170848364266136042527,1276022762028643136592313,14225223924067129319855681
%N A336789 Incrementally largest values of minimal y satisfying the equation x^2 - D*y^2 = 2, where D is a prime number.
%e A336789 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 A336789 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 A336789 From _Jon E. Schoenfield_, Feb 24 2021: (Start)
%e A336789 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 A336789 .
%e A336789          x values satisfying     minimal
%e A336789     D      x^2 - D*y^2 = 2       y value  record
%e A336789   ---  ------------------------  -------  ------
%e A336789     2  1, 7, 41, 239, 1393, ...      1       *
%e A336789     7  1, 17, 271, 4319, ...         1
%e A336789    23  1, 49, 2351, 112799, ...      1
%e A336789    31  7, 21287, 64712473, ...       7       *
%e A336789    47  1, 97, 9311, 893759, ...      1
%e A336789    71  7, 48727, 339139913, ...      7
%e A336789    79  1, 161, 25759, ...            1
%e A336789   103  47, 21387679, ...            47       *
%e A336789   ...
%e A336789 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 A336789 Cf. A033315, A336788.
%K A336789 nonn
%O A336789 1,2
%A A336789 _Christine Patterson_, Oct 14 2020
%E A336789 Example section edited by _Jon E. Schoenfield_, Feb 24 2021