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 A033313 Smallest positive integer x satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D and positive y. 25
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified May 18 02:04 EDT 2024. Contains 372615 sequences. (Running on oeis4.)