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A033313
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Smallest positive integer x satisfying the Pell equation x^2 - D*y^2 = 1 for nonsquare D and positive y.
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24
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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
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OFFSET
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1,1
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LINKS
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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]
H. W. Lenstra, jr., Solving the Pell Equation
F. Richman & 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
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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r[n_] := Reduce[x > 0 && y > 0 && x^2 - n*y^2 == 1, {x, y}, Integers] /. C[_] -> k; sol[n_] := Catch[For[k = 0, True, k++, rn = r[n]; If[rn =!= False, Throw[rn]]]]; A033313 = Reap[For[n = 2, n <= 70, n++, If[! IntegerQ[Sqrt[n]], Sow[x /. ToRules[sol[n]]]]]][[2, 1]](* Jean-François Alcover, Apr 25 2012 *)
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CROSSREFS
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See A033317 (for y's).
Cf. A000037, A002350.
Sequence in context: A064614 A234747 A016650 * A231442 A319107 A228323
Adjacent sequences: A033310 A033311 A033312 * A033314 A033315 A033316
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KEYWORD
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nonn
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AUTHOR
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Eric W. Weisstein
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EXTENSIONS
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Offset switched to 1 by R. J. Mathar, Sep 21 2009
Name corrected by Wolfdieter Lang, Sep 03 2015
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STATUS
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approved
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