A033313 Smallest positive integer x satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D and positive y.

3, 2, 9, 5, 8, 3, 19, 10, 7, 649, 15, 4, 33, 17, 170, 9, 55, 197, 24, 5, 51, 26, 127, 9801, 11, 1520, 17, 23, 35, 6, 73, 37, 25, 19, 2049, 13, 3482, 199, 161, 24335, 48, 7, 99, 50, 649, 66249, 485, 89, 15, 151, 19603, 530, 31, 1766319049, 63, 8, 129, 65, 48842, 33

Offset

1,1

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

S. R. Finch, Class number theory [Cached copy, with permission of the author]

Bernard Frénicle de Bessy, Solutio duorum problematum circa numeros cubos et quadratos, (1657). Bibliothèque Nationale de Paris. See column C page 19.

H. W. Lenstra, jr., Solving the Pell Equation, Notices of the AMS, Vol.49, No.2, Feb. 2002, p.182-192.

F. Richman and R. Mines, Pell's equation

Derek Smith, Historical Overview of Pell Equations

Derek Smith, The Search For An Exhaustive Solution to Pell's Equation

Eric Weisstein's World of Mathematics, Pell Equation

Formula

a(n) = sqrt(1 + A000037(n)*A033317(n)^2), or

a(n) = sqrt(1 + (n + floor(1/2 + sqrt(n)))*A033317(n)^2). - Zak Seidov, Oct 24 2013

Maple

F:= proc(d) local r, Q; uses numtheory;
  Q:= cfrac(sqrt(d), 'periodic', 'quotients'):
  r:= nops(Q[2]);
  if r::odd then
    numer(cfrac([op(Q[1]), op(Q[2]), op(Q[2][1..-2])]))
  else
    numer(cfrac([op(Q[1]), op(Q[2][1..-2])]));
  fi
end proc:
map(F, remove(issqr, [$1..100])); # Robert Israel, May 17 2015

Mathematica

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[Sqrt[m]]; n = Length[Last[cf]]; If[n == 0, Return[{}]]; If[OddQ[n], n = 2n]; s = FromContinuedFraction[ContinuedFraction[Sqrt[m], n]]; {Numerator[s], Denominator[s]}];
A033313 = DeleteCases[PellSolve /@ Range[100], {}][[All, 1]] (* Jean-François Alcover, Nov 21 2020, after N. J. A. Sloane in A002350 *)
Table[If[! IntegerQ[Sqrt[k]], {k, FindInstance[x^2 - k*y^2 == 1 && x > 0 && y > 0, {x, y}, Integers]}, Nothing], {k, 2, 80}][[All, 2, 1, 1, 2]] (* Horst H. Manninger, Mar 28 2021 *)

Crossrefs

See A033317 (for y's).

Cf. A000037, A002350, A077232, A077233.

Sequence in context: A064614 A234747 A016650 * A231442 A319107 A228323

Adjacent sequences: A033310 A033311 A033312 * A033314 A033315 A033316

Keyword

nonn

Author

Eric W. Weisstein

Extensions

Offset switched to 1 by R. J. Mathar, Sep 21 2009

Name corrected by Wolfdieter Lang, Sep 03 2015

Status

approved