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A077426
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Nonsquare integers n such that the continued fraction (sqrt(n)+1)/2 has odd period length.
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10
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5, 13, 17, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 173, 181, 185, 193, 197, 229, 233, 241, 257, 265, 269, 277, 281, 293, 313, 317, 325, 337, 349, 353, 365, 373, 389, 397, 401, 409, 421, 425, 433, 445, 449, 457, 461, 481, 485
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OFFSET
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1,1
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COMMENTS
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Nonsquare integers n for which Pell equation x^2 - n*y^2 = -4 has infinitely many integer solutions. The smallest solutions are given in A078356 and A078357.
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REFERENCES
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O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, table p. 108).
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LINKS
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MAPLE
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isOddPrim := proc(n::integer)
local cf;
cf := numtheory[cfrac]((sqrt(n)+1)/2, 'periodic', 'quotients') ;
if nops(op(2, cf)) mod 2 =1 then
RETURN(true) ;
else
RETURN(false) ;
fi ;
end:
notA077426 := proc(n::integer)
if issqr(n) then
RETURN(true) ;
elif not isOddPrim(n) then
RETURN(true) ;
else
RETURN(false) ;
fi ;
end:
local resul, i ;
resul := 5 ;
i := 1 ;
while i < n do
resul := resul+4 ;
while notA077426(resul) do
resul := resul+4 ;
od ;
i:= i+1 ;
od ;
RETURN(resul) ;
end:
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MATHEMATICA
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fQ[n_] := !IntegerQ@ Sqrt@ n && OddQ@ Length@ ContinuedFraction[(Sqrt@ n + 1)/2][[2]]; Select[Range@ 500, fQ] (* Robert G. Wilson v, Nov 17 2012 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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