cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A336792 Values of odd prime numbers, D, for incrementally largest values of minimal positive y satisfying the equation x^2 - D*y^2 = -2.

Original entry on oeis.org

3, 19, 43, 67, 139, 211, 331, 379, 571, 739, 859, 1051, 1291, 1531, 1579, 1699, 2011, 2731, 3019, 3259, 3691, 3931, 5419, 5659, 5779, 6211, 6379, 6451, 8779, 9619, 10651, 16699, 17851, 18379, 21739, 25939, 32971, 42331, 42571, 44851, 50131, 53299, 55819, 56611, 60811, 61051, 73459, 76651, 90619, 90931
Offset: 1

Views

Author

Christine Patterson, Oct 14 2020

Keywords

Comments

For the corresponding y values see A336793.
For solutions of this Diophantine equation it is sufficient to consider the odd primes p(n) := A007520(n), for n >= 1, the primes 3 (mod 8). Also prime 2 has the fundamental solution (x, y) = (0, 1). If there is a solution for p(n) then there is only one infinite family of solutions because there is only one representative parallel primitive binary quadratic form for Discriminant Disc = 4*p(n) and representation k = -2. Only proper solutions can occur. The conjecture is that each p(n) leads to solutions. For the fundamental solutions (with prime 2) see A339881 and A339882. - Wolfdieter Lang, Dec 22 2020

Examples

			For D=3, the least positive y for which x^2-D*y^2=-2 has a solution is 1. The next prime, D, for which x^2-D*y^2=-2 has a solution is 11, but the smallest positive y in this case is also 1, which is equal to the previous record y. So 11 is not a term.
The next prime, D, after 11 for which x^2-D*y^2=-2 has a solution is 19 and the least positive y for which it has a solution is y=3, which is larger than 1, so it is a new record y value. So 19 is a term of this sequence and 3 is a term of A336793.

Links

Crossrefs

Cf. A033316 (analogous for x^2-D*y^2=1), A336790 (similar sequence for x's), A336793.

A336791 Incrementally largest values of minimal x satisfying the equation x^2 - D*y^2 = -2, where D is an odd prime number.

Original entry on oeis.org

1, 3, 13, 59, 221, 8807, 527593, 52778687, 113759383, 13458244873, 313074529583, 1434867510253, 30909266676193, 842239594152347, 1075672117707143, 29204057639975683, 52376951398984393, 4785745078256208692917, 15280437983663153103594943
Offset: 1

Views

Author

Christine Patterson, Oct 14 2020

Keywords

Comments

Analogous to A033315 for x^2-D*y^2=1, and D required to be prime.

Examples

			For D=43, the least x for which x^2-D*y^2=-2 has a solution is 59. The next prime, D, for which x^2-D*y^2=-2 has a solution is 59, but the smallest x in this case is 23, which is less than 59. The next prime, D, after 59 for which x^2-D*y^2=-2 has a solution is 67 and the least x for which it has a solution is 221, which is larger than 59, so it is a new record value. 67 is a term of A336790 and 221 is a term of this sequence, but 59 is not a term of A336790 because the least x for which x^2-47*y^2=-2 has a solution at D=59 is not a record value.

Links

Crossrefs

Cf. A033315, A336790.

Programs

  • Mathematica
    records[n_]:=Module[{ri=n,m=0,rcs={},len},len=Length[ri];While[ len>0,If[ First[ri]>m,m=First[ri];AppendTo[rcs,m]]; ri=Rest[ri]; len--];rcs]; records[ Abs[Flatten[Table[x/.FindInstance[x^2-p y^2==-2,{x,y},Integers],{p,Prime[Range[2,500]]}]/.x->Nothing]]] (* Harvey P. Dale, Jan 02 2022 *)

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

Views

Author

Christine Patterson, Jan 17 2021

Keywords

Comments

Is 61 the only term where this differs from A336794? - R. J. Mathar, Feb 16 2021

Examples

			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.

Links

Crossrefs

Cf. A033316 (analog for x^2-D*y^2=1), A336790 (similar sequence for x's), A336800, A336794.
Showing 1-3 of 3 results.

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