A341084
Incrementally largest values of minimal x satisfying the equation x^2 - D*y^2 = -5, where D is a prime number.
Original entry on oeis.org
0, 16, 164, 1061372, 103068308, 162122886, 123398206659664, 2466743672871107188, 36438755210133838109283894464, 1957006192940494702014893262914, 541745559127518723115014358590896, 83890612389598737813497437560727166
Offset: 1
For D=29, the least x for which x^2 - D*y^2 = -5 has a solution is 16. The next prime, D, for which x^2 - D*y^2 = -5 has a solution is 41, but the smallest x in this case is 6, which is less than 16. The next prime, D, after 41 for which x^2 - D*y^2 = -5 has a solution is 61 and the least x for which it has a solution is 164, which is larger than 16, so it is a new record value. 29 is a term of A341083 and 16 is a term of this sequence, but 41 is not a term of A341083 because the least x for which x^2 - D*y^2 = -5 has a solution is not a record value.
A341085
Value of prime number D for incrementally largest values of minimal y satisfying the equation x^2 - D*y^2 = -5.
Original entry on oeis.org
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
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)
Showing 1-2 of 2 results.
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