A336786
Value of prime number D for incrementally largest values of minimal x satisfying the equation x^2 - D*y^2 = 2.
Original entry on oeis.org
2, 7, 23, 31, 71, 103, 127, 151, 199, 271, 463, 631, 751, 919, 991, 1471, 1759, 1831, 1999, 2311, 2671, 3319, 4111, 4519, 4951, 5119, 6679, 8191, 8719, 10399, 11839, 12919, 13399, 15031, 16879, 19231, 21319, 23599, 26959, 30319, 32839, 34519, 37591, 38119, 43759, 48799
Offset: 1
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, so it is a new record value. 71 is a term of this sequence and 59 is a term of A336787. 47 is not a term here because the least x for which x^2-47y^2=2 has a solution is not a record value.
A336796
Value of prime number D for incrementally largest values of minimal y satisfying the equation x^2 - D*y^2 = 3.
Original entry on oeis.org
13, 73, 109, 157, 241, 277, 421, 1549, 3061, 4561, 4861, 5701, 6301, 6829, 8941, 10429, 13381, 14029, 14221, 21169, 22369, 24049, 26161, 29761, 30529, 33601, 39901, 44221, 45061, 47581, 55609, 61609, 62869, 64381, 74869, 97549, 121501, 129061, 133669, 135661
Offset: 1
For D=13, the least positive y for which x^2-D*y^2=3 has a solution is 1. The next prime, D, for which x^2-D*y^2=3 has a solution is 61, but the smallest positive y in this case is also 1, which is equal to the previous record y. So, 61 is not a term.
The next prime, D, after 61 for which x^2-D*y^2=3 has a solution is 73, and the least positive y for which it has a solution in this case is y=11, which is larger than 1, so it is a new record y value. So, 73 is a term in this sequence and 11 is a term in A336800.
A341077
Value of prime number D for incrementally largest values of minimal y satisfying the equation x^2 - D*y^2 = -3.
Original entry on oeis.org
3, 13, 61, 181, 397, 541, 661, 1021, 1381, 1621, 3361, 3529, 4201, 4261, 4621, 6421, 9241, 9601, 9949, 12541, 20161, 23209, 25309, 32869, 37321, 43261, 71821, 78901, 82021, 112429, 127261, 131041, 137089, 139309, 144169, 169789, 183661, 226669, 300301
Offset: 1
For D=13, the least positive y for which x^2 - D*y^2 = -3 has a solution is 2. The next primes, D, for which x^2 - D*y^2 = -3 has a solution are 19, 31, and 43, but the smallest positive y in each of those cases is 1 or 2, neither of which is larger than the previous record y, 2. So 19, 31, and 43 are not terms of this sequence.
The next prime, D, after 43 for which x^2 - D*y^2 = -3 has a solution is 61, and the least positive y for which it has a solution is y=722, which is larger than 2, so it is a new record y value. So 61 is a term of this sequence and 722 is the corresponding term of A341078.
A341081
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
19, 61, 149, 241, 409, 421, 541, 1069, 1249, 1381, 1621, 4261, 4621, 4789, 6301, 8269, 12601, 12721, 14449, 16069, 20101, 32029, 33889, 34381, 35281, 38329, 43261, 45061, 60589, 87481, 89989, 97549, 99661, 121081, 125101, 166021, 178621, 187069, 191689, 202381
Offset: 1
For D=19, the least positive y for which x^2-D*y^2=5 has a solution is 2. The next prime, D, for which x^2-D*y^2=5 has a solution is 29, but the smallest positive y in this case is 2, which is equal to the previous record y. So, 29 is not a term.
The next prime, D, after 19 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=58, which is larger than 2, so it is a new record y value. So 61 is in this sequence and 58 is in A341082.
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)
A336788
Values of prime numbers, D, for incrementally largest values of minimal y satisfying the equation x^2 - D*y^2 = 2.
Original entry on oeis.org
2, 31, 103, 127, 151, 199, 271, 463, 631, 751, 919, 991, 1471, 1759, 1831, 1999, 2311, 2671, 3319, 4111, 4519, 4951, 5119, 6679, 8191, 8719, 10399, 11839, 12919, 13399, 15031, 16879, 19231, 21319, 23599, 26959, 30319, 32839, 34519, 37591, 38119, 43759, 48799, 53551, 58111, 62791
Offset: 1
For D = 2, the least y for which x^2 - D*y^2 = 2 has a solution is 1.
The next primes, D, for which x^2 - D*y^2 = 2 has a solution are 7 and 23, but the smallest y in each of these cases is also 1, which is equal to the previous record y. So neither 7 nor 23 is a term.
The next prime, D, after 23 for which x^2 - D*y^2 = 2 has a solution is 31 and the least y for which it has a solution there is y = 7, which is larger than 1, so it is a new record y value. So 31 is a term here, and 7 is the corresponding term of A336789.
A341087
Value of prime number D for incrementally largest values of minimal x satisfying the equation x^2 - D*y^2 = 6.
Original entry on oeis.org
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
Offset: 1
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)
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