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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
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).
Sequence in context: A064614 A234747 A016650 * A231442 A319107 A228323
KEYWORD
nonn
AUTHOR
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.)