A341085 Value of prime number D for incrementally largest values of minimal y satisfying the equation x^2 - D*y^2 = -5.

5, 29, 61, 109, 181, 661, 1021, 1549, 2161, 2389, 3169, 3469, 4909, 5581, 8929, 9601, 9949, 12841, 13381, 14029, 17029, 21169, 24709, 25309, 28729, 31249, 32869, 34549, 35149, 39901, 40429, 43801, 48049, 49009, 56401, 56701, 62701, 63541, 70141, 86269

Offset

1,1

Links

Table of n, a(n) for n=1..40.

Christine Patterson, COCALC (Sage) Program

Example

For D=29, the least positive y for which x^2 - D*y^2 = -5 has a solution is 3. The next prime, D, for which x^2 - D*y^2 = -5 has a solution is 41, but the smallest positive y in this case is 1, which is less than the previous record y, 3. So, 41 is not a term.
The next prime, D, after 41 for which x^2 - D*y^2 = -5 has a solution is 61 and the least positive y for which it has a solution is y=21, which is larger than 3, so it is a new record y value. So 61 is a term of this sequence and 21 is the corresponding term of A341086.
From Jon E. Schoenfield, Feb 20 2021: (Start)
As D runs through the primes, the minimal y values satisfying the equation x^2 - D*y^2 = -5 begin as follows:
.
      y values satisfying   minimal
   D    x^2 - D*y^2 = -5    y value  record
  --  --------------------  -------  ------
   2  (none)
   3  (none)
   5  1, 9, 161, 2889, ...     1       *
   7  (none)
  11  (none)
  13  (none)
  17  (none)
  19  (none)
  23  (none)
  29  3, 283, 58523, ...       3       *
  31  (none)
  37  (none)
  41  1, 129, 3969, ...        1
  43  (none)
  47  (none)
  51  (none)
  53  (none)
  59  (none)
  61  21, 3447309, ...        21       *
  ...
The record high minimal values of y (marked with asterisks) are the terms of A341086. The corresponding values of D are the terms of this sequence. (End)

Crossrefs

Cf. A033316 (analogous for x^2 - D*y^2 = 1), A341083 (similar sequence for x's), A341086.

Sequence in context: A031394 A103094 A341083 * A301858 A293174 A108928

Adjacent sequences: A341082 A341083 A341084 * A341086 A341087 A341088

Keyword

nonn

Author

Christine Patterson, Feb 13 2021

Extensions

a(1)=5 inserted by Jon E. Schoenfield, Feb 20 2021

Status

approved