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

Author

Eric W. Weisstein

Keywords

• nonn

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

Crossrefs

See A033317 (for y's).

Cf. A000037, A002350, A077232, A077233.

Programs

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 *)

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

Extensions

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

Name corrected by Wolfdieter Lang, Sep 03 2015