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A341087 Value of prime number D for incrementally largest values of minimal x satisfying the equation x^2 - D*y^2 = 6. 1
3, 19, 43, 67, 139, 211, 571, 691, 883, 1483, 2011, 2539, 2851, 3331, 3931, 5779, 8011, 8779, 9811, 10459, 11131, 17851, 18379, 33331, 34819, 38299, 42571, 56659, 62731, 65179, 79699, 90931, 91939, 93811, 95419, 102859, 130579, 138139, 170179, 196771, 204019, 223939, 234259, 254731, 285139 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Analogous to A033316 for x^2 - D*y^2 = 1, and D is required to be prime, and for record values of x.

LINKS

Table of n, a(n) for n=1..45.

Christine Patterson, COCALC (Sage) Program

EXAMPLE

For D=139, the least x for which x^2 - D*y^2 = 6 has a solution is 59. The next prime, D, for which x^2 - D*y^2 = 6 has a solution is 163, but the smallest x in this case is 13, which is less than 59. The next prime, D, after 163 for which x^2 - D*y^2 = 6 has a solution is 211 and the least x for which it has a solution is 27265, which is larger than 59, so it is a new record value. So 139 is a term of this sequence and 59 is the corresponding term of A341088, but 163 is not a term here because the least x for which x^2 - D*y^2 = 6 has a solution is not a record value.

From Jon E. Schoenfield, Feb 20 2021: (Start)

As D runs through the primes, the minimal x values satisfying the equation x^2 - D*y^2 = 6 begin as follows (with primes D for which there are no solutions omitted):

.

       x values satisfying   minimal

    D    x^2 - D*y^2 = 6     x value  record

   --  --------------------  -------  ------

    3  3, 9, 33, 123, ...         3     *

   19  5, 109, 1591, ...          5     *

   43  7, 1541, 47207, ...        7     *

   67  41, 3577, ...             41     *

  139  59, 3945595, ...          59     *

  163  13, 14921333, ...         13

  211  27265, 30627659, ...   27265     *

  ...

The record high minimal values of x (marked with asterisks) are the terms of A341088. The corresponding values of D are the terms of this sequence. (End)

CROSSREFS

Cf. A033316, A341088.

Sequence in context: A107154 A141373 A336792 * A031393 A146672 A146704

Adjacent sequences:  A341084 A341085 A341086 * A341095 A341096 A341097

KEYWORD

nonn

AUTHOR

Christine Patterson, Feb 13 2021

EXTENSIONS

a(1), a(2) inserted by Jon E. Schoenfield, Feb 20 2021

STATUS

approved

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Last modified October 20 03:04 EDT 2021. Contains 348099 sequences. (Running on oeis4.)