A336787 Incrementally largest values of minimal x satisfying the equation x^2 - D*y^2 = 2, where D is a prime number.

2, 3, 5, 39, 59, 477, 2175, 41571, 127539, 340551, 15732537, 221272626669, 2700614460969, 66944775830061, 616049024759241, 6245844517335369, 13085071811371140879, 43795350588094552821, 63464174140920940599, 633160367499665048108061

Offset

1,1

Comments

Analogous to A033315 for x^2 - D*y^2 = 1, and D required to be prime. The x values incrementally largest for x^2 - D*y^2 = 2. D values appear in sequence A336786.

Links

Table of n, a(n) for n=1..20.

Christine Patterson, Sage Program

Example

For D=31, the least x for which x^2 - Dy^2 = 2 has a solution is 39. The next prime, D, for which x^2 - Dy^2 = 2 has a solution is 47, but the smallest x in this case is 7, which is less than 39. The next prime, D, after 47 for which x^2 - Dy^2 = 2 has a solution is 71 and the least x for which it has a solution is x=59, which is larger than 39, a new record value, so 71 is a term of A336786 and 59 is the corresponding term of this sequence. 47 is not a term of A336786 because the least x for which x^2 - 47*y^2 = 2 has a solution is not a record value.
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 x values satisfying the equation x^2 - D*y^2 = 2 begin as follows:
.
           x values satisfying      minimal
    D        x^2 - D*y^2 = 2        x value  record
  ---  ---------------------------  -------  ------
    2  2, 10, 58, 338, 1970, ...        2      *
    7  3, 45, 717, 11427, ...           3      *
   23  5, 235, 11275, 540965, ...       5      *
   31  39, 118521, 360303801, ...      39      *
   47  7, 665, 63833, 6127303, ...      7
   71  59, 410581, 2857643701, ...     59      *
   79  9, 1431, 228951, ...             9
  103  477, 217061235, ...            477      *
  ...
The record high minimal values of x (marked with asterisks) are the terms of this sequence. The corresponding values of D are the terms of A336786. (End)

Crossrefs

Cf. A033315, A336786.

Sequence in context: A347720 A128026 A041443 * A057775 A215321 A215309

Adjacent sequences: A336784 A336785 A336786 * A336788 A336789 A336790

Keyword

nonn

Author

Christine Patterson, Aug 05 2020

Extensions

a(1)=2 inserted and Example section edited by Jon E. Schoenfield, Feb 24 2021

Status

approved