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
Keywords
Examples
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)
Links
- Christine Patterson, COCALC (Sage) Program
Extensions
a(1)=5 inserted by Jon E. Schoenfield, Feb 20 2021