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A107997 Squarefree integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y odd.

%I A107997 #36 Aug 11 2025 07:10:33
%S A107997 5,13,21,29,53,61,69,77,85,93,109,133,149,157,165,173,181,205,213,221,
%T A107997 229,237,253,277,285,293,301,309,317,341,357,365,397,413,421,429,437,
%U A107997 445,453,461,469,493,501,509,517,533,541,565,581,589,597,613,629,645
%N A107997 Squarefree integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y odd.
%C A107997 Squarefree integers m for which the fundamental unit of Q(sqrt(m)) is of the form (u + v*sqrt(m))/2, where u and v are both odd.
%D A107997 E. L. Ince, Cycles of Reduced Ideals in Quadratic Fields, British Association Mathematical Tables, Vol. IV, London, 1934.
%D A107997 H. C. Williams, Eisenstein's problem and continued fractions, Utilitas Math. 37 (1990) 145-157.
%H A107997 Robert G. Wilson v, <a href="/A107997/b107997.txt">Table of n, a(n) for n = 1..1000</a> (first 161 terms from Charles R Greathouse IV)
%H A107997 F. Arndt, <a href="https://www.digitale-sammlungen.de/en/view/bsb10593875?page=524,525">Beiträge zur Theorie der quadratischen Formen</a>, Archiv der Mathematik und Physik 15 (1850) 467-478.
%H A107997 A. Cayley, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0053">Note sur l'équation x^2 - D*y^2 = +-4, D=5 (mod 8)</a>, J. Reine Angew. Math. 53 (1857) 369-371.
%H A107997 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Class number theory</a>
%H A107997 Steven R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a> [Cached copy, with permission of the author]
%H A107997 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FundamentalUnit.html">Fundamental unit</a>
%t A107997 fQ[n_] := Block[{nffu = NumberFieldFundamentalUnits@ Sqrt@ n}, SquareFreeQ@ n && Denominator[ nffu[[1, 2, 2]]] > 1]; Select[ 8Range@ 81 - 3, fQ] (* _Robert G. Wilson v_, Dec 22 2014 *)
%Y A107997 Cf. A107998.
%K A107997 nonn
%O A107997 1,1
%A A107997 _Steven Finch_, Jun 13 2005