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Nonsquare positive integers k such that the fundamental unit of the quadratic field Q(sqrt(k)) has norm -1 and can be written as x + y*sqrt(d) with integers x, y where d is the squarefree part of k.
8

%I #17 Sep 04 2023 15:48:50

%S 2,8,10,17,18,26,32,37,40,41,50,58,65,68,72,73,74,82,89,90,97,98,101,

%T 104,106,113,122,128,130,137,145,148,153,160,162,164,170,185,193,197,

%U 200,202,218,226,232,233,234,241,242,250,257,260,265,269,272,274

%N Nonsquare positive integers k such that the fundamental unit of the quadratic field Q(sqrt(k)) has norm -1 and can be written as x + y*sqrt(d) with integers x, y where d is the squarefree part of k.

%C This sequence is a subsequence of A172000.

%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, 400}]; cr

%Y Cf. A087643, A172000, A194366.

%K nonn

%O 1,1

%A _Artur Jasinski_, Oct 10 2011

%E Definition clarified by _Emmanuel Vantieghem_, Mar 06 2017