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Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(d))has norm -1 and minimum one from two parts of fundamental unit are not integer.
0

%I #6 Mar 31 2012 10:22:18

%S 5,13,20,29,45,52,53,61,80,85,109,116,117,125,149,157,173,180,181,208,

%T 212,229,244,245,261,277,293,317,320,325,340,365,397,405,421,436,445,

%U 461,464,468,477,493,500,509,533,541,549,565,596,605,613,628,629,637

%N Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(d))has norm -1 and minimum one from two parts of fundamental unit are not integer.

%C Numbers which occured in A172000 and not in A197115.

%t cr = {}; Do[ If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d3 = Expand[(d1 + d2) (d1 - d2)]; If[d3 == -1, k1 = Max[Denominator[d1], Denominator[d2]]; If[k1 == 1, , AppendTo[cr, n]]]], {n, 2, 2000}]; cr

%Y Cf. A087643, A172000, A194366, A197115.

%K nonn

%O 1,1

%A _Artur Jasinski_, Oct 10 2011

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