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A298212
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Smallest n such that A060645(a(n)) = 0 (mod n), i.e., x=A023039(a(n)) and y=A060645(a(n)) is the fundamental solution of the Pell equation x^2 - 5*(n*y)^2 = 1.
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3
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1, 1, 2, 1, 5, 2, 4, 2, 2, 5, 5, 2, 7, 4, 10, 4, 3, 2, 3, 5, 4, 5, 4, 2, 25, 7, 6, 4, 7, 10, 5, 8, 10, 3, 20, 2, 19, 3, 14, 10, 10, 4, 22, 5, 10, 4, 8, 4, 28, 25, 6, 7, 9, 6, 5, 4, 6, 7, 29, 10, 5, 5, 4, 16, 35, 10, 34, 3, 4, 20, 35, 2, 37, 19, 50
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OFFSET
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1,3
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COMMENTS
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The fundamental solution of the Pell equation x^2 - 5*(n*y)^2 = 1, is the smallest solution of x^2 - 5*y^2 = 1 satisfying y = 0 (mod n).
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REFERENCES
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Michael J. Jacobson, Jr. and Hugh C. Williams, Solving the Pell Equation, Springer, 2009, pages 1-17.
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LINKS
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FORMULA
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a(n) <= n.
if n | m then a(n) | a(m).
a(5^m) = 5^m for m>=0.
In general: if p is prime and p = 1 (mod 4) then: a(n) = n iff n = p^m, for m>=0.
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MATHEMATICA
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b[n_] := b[n] = Switch[n, 0, 0, 1, 4, _, 18 b[n - 1] - b[n - 2]];
a[n_] := For[k = 1, True, k++, If[Mod[b[k], n] == 0, Return[k]]];
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PROG
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(Python)
xf, yf = 9, 4
x, n = 2*xf, 0
while n < 20000:
n = n+1
y1, y0, i = 0, yf, 1
while y0%n != 0:
y1, y0, i = y0, x*y0-y1, i+1
print(n, i)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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