A033317 Smallest positive integer y satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D.
2, 1, 4, 2, 3, 1, 6, 3, 2, 180, 4, 1, 8, 4, 39, 2, 12, 42, 5, 1, 10, 5, 24, 1820, 2, 273, 3, 4, 6, 1, 12, 6, 4, 3, 320, 2, 531, 30, 24, 3588, 7, 1, 14, 7, 90, 9100, 66, 12, 2, 20, 2574, 69, 4, 226153980, 8, 1, 16, 8, 5967, 4, 936, 30, 413, 2, 267000, 430, 3, 6630, 40, 6, 9
Offset: 1
Keywords
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
- Laurent Beeckmans, Squares Expressible as Sum of Consecutive Squares, Am. Math. Monthly, Volume 101, Number 5, page 442, May 1994.
- 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 B page 19.
- N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
- Eric Weisstein's World of Mathematics, Pell Equation
Programs
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Maple
F:= proc(d) local r,Q; uses numtheory; Q:= cfrac(sqrt(d),'periodic','quotients'): r:= nops(Q[2]); if r::odd then denom(cfrac([op(Q[1]),op(Q[2]),op(Q[2][1..-2])])) else denom(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]}]; A033317 = DeleteCases[PellSolve /@ Range[100], {}][[All, 2]] (* Jean-François Alcover, Nov 21 2020, after N. J. A. Sloane in A002349 *)
Comments