A341078
Incrementally largest values of minimal y satisfying the equation x^2 - D*y^2 = -3, where D is a prime number.
Original entry on oeis.org
1, 2, 722, 837158, 77228318, 5436980738, 49637737974482, 462761120757722506058, 2836540596515452087502, 37216095020093890760397134162, 1858485134141860820807351059562927114738, 42507485681147639763501995374671391449914
Offset: 1
From _Jon E. Schoenfield_, Feb 23 2021: (Start)
As D runs through the primes, the minimal y values satisfying the equation x^2 - D*y^2 = -3 begin as follows:
.
x values satisfying minimal
D x^2 - D*y^2 = -5 y value record
-- ---------------------- ------- ------
2 (none)
3 1, 2, 7, 26, 97, ... 1 *
5 (none)
7 1, 2, 14, 31, 223, ... 1
11 (none)
13 2, 38, 2558, ... 2 *
17 (none)
19 1, 14, 326, 4759, ... 1
23 (none)
29 (none)
31 2, 37, 604, ... 2
37 (none)
41 (none)
43 2, 61, 13867, ... 2
47 (none)
53 (none)
59 (none)
61 722, 60158, ... 722 *
...
The record high minimal values of y (marked with asterisks) are the terms of this sequence. The corresponding values of D are the terms of A341077. (End)
A341080
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
9, 11, 13, 453, 23461, 544557, 1537329309, 23841388917, 5420031851795067, 187413651300546981, 217796221885036092531, 177582465273740054778830373, 160849509983404119454318443146043, 608375445734704350836734541937669395740416570597
Offset: 1
For D=29, the least x for which x^2 - D*y^2 = 5 has a solution is 11. The next prime, D, for which x^2 - D*y^2 = 5 has a solution is 31, but the smallest x in this case is 6, which is less than 11. The next prime, D, after 31 for which x^2 - D*y^2 = 5 has a solution is 41 and the least x for which it has a solution is 13, which is larger than 11, so it is a new record value. 29 is a term of A341079 and 11 is a term of this sequence, but 31 is not a term of A341079 because the least x for which x^2 - D*y^2 = 5 has a solution is not a record value.
From _Jon E. Schoenfield_, Feb 18 2021: (Start)
As D runs through the primes, the minimal x values satisfying the equation x^2 - D*y^2 = 5 begin as follows:
.
x values minimal
D satisfying x^2 - D*y^2 = 5 x value record
-- -------------------------- ------- ------
2 (none)
3 (none)
5 5, 85, 1525, 27365, ... 5 *
7 (none)
11 4, 7, 73, 136, 1456, ... 4
13 (none)
17 (none)
19 9, 48, 3012, 16311, ... 9 *
29 11, 2251, 213371, ... 11 *
31 6, 657, 17583, ... 6
41 13, 397, 52877, ... 13 *
59 8, 169, 8311, 179132, ... 8
61 453, 9747957, ... 453 *
...
The record high values of x (marked with asterisks) are the terms of this sequence. The corresponding values of D are the terms of A341079. (End)
A341082
Incrementally largest values of minimal y satisfying the equation x^2-D*y^2=5, where D is a prime number.
Original entry on oeis.org
2, 58, 1922, 35078, 76016042, 1161958198, 233025369988282, 5732081667022982, 6162672978871449862, 4778628197827994122556402, 3995105338251652225860073210642, 9319999956851141533879334192705803394284705042
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 a term of A341081 and 58 is a term of this sequence.
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.
A341086
Incrementally largest values of minimal y satisfying the equation x^2 - D*y^2 = -5, where D is a prime number.
Original entry on oeis.org
1, 3, 21, 101661, 7661007, 4799633969721, 77198907060727563, 925844015429395821936018843, 42098324998788084039841633029, 11083764383781783138639570812583, 1490226373435897063030119543467763
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 A341085 and 21 is a term of this sequence.
A336789
Incrementally largest values of minimal y satisfying the equation x^2 - D*y^2 = 2, where D is a prime number.
Original entry on oeis.org
1, 7, 47, 193, 3383, 9041, 20687, 731153, 8808724183, 98546821297, 2208304390649, 19569442212887, 162848901149273, 311991807873328639, 1023490545293318137, 1419456983764900351, 13170848364266136042527, 1276022762028643136592313, 14225223924067129319855681
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 7 and 23 are not terms of A336788.
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 of A336788, and 7 is the corresponding term here.
From _Jon E. Schoenfield_, Feb 24 2021: (Start)
Primes D for which the equation x^2 - D*y^2 = 2 has integer solutions begin 2, 7, 23, 31, 47, 71, 79, 103, ...; at those values of D, the minimal y values satisfying the equation x^2 - D*y^2 = 2 begin as follows:
.
x values satisfying minimal
D x^2 - D*y^2 = 2 y value record
--- ------------------------ ------- ------
2 1, 7, 41, 239, 1393, ... 1 *
7 1, 17, 271, 4319, ... 1
23 1, 49, 2351, 112799, ... 1
31 7, 21287, 64712473, ... 7 *
47 1, 97, 9311, 893759, ... 1
71 7, 48727, 339139913, ... 7
79 1, 161, 25759, ... 1
103 47, 21387679, ... 47 *
...
The record high minimal values of y (marked with asterisks) are the terms of this sequence. The corresponding values of D are the terms of A336788. (End)
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