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%I #33 Jan 16 2023 04:23:03
%S 5,13,21,29,45,53,61,69,77,85,93,109,117,125,133,149,157,165,173,181,
%T 205,213,221,229,237,245,253,261,277,285,293,301,309,317,341,357,365,
%U 397,413,421,429,437,445,453,461,469,477,493,501,509,517,525,533,541
%N 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 From _Wolfdieter Lang_, Oct 30 2015: (Start)
%C These numbers m are the members of A079896 that have two conjugacy classes of proper solutions (and one of improper solutions) for the Pell equation x^2 - m*y^2 = +4. E.g., m = 5 has the proper positive fundamental solutions (3,1) and (7,3) obtained from (3,-1) (and the improper positive fundamental solution (18,8) = 2*(9,4) obtained from (2,0)).
%C For these numbers m one has therefore two conjugacy classes of improper solutions, and, in addition, the improper ambiguous class with member (4, 0) for the equation X^2 - m*Y^2 = +16.
%C Note that also even m may have solutions with both x and y odd, e.g., m = 12 with minimal positive solution (x, y) = (4, 1) for the +4 equation. The +-4 in the name means +4 or -4 (inclusive).
%C (End)
%H 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 A. Cayley, <a href="https://www.digizeitschriften.de/dms/img/?PID=GDZPPN002149834">Note sur l'équation x^2 - D*y^2 = +-4, D=5 (mod. 8)</a>, J. Reine Angew. Math. 53 (1857) 369-371.
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Class number theory</a>
%H Steven R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a> [Cached copy, with permission of the author]
%H N. Ishii, P. Kaplan and K. S. Williams, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa54/aa5446.pdf">On Eisenstein's problem</a>, Acta Arith. 54 (1990) 323-345.
%H Wolfdieter Lang, <a href="/A225953/a225953_3.pdf">Periods of Indefinite Binary Quadratic Forms, Continued Fractions and the Pell +/-4 Equations</a>.
%Y Cf. A079896.
%K nonn
%O 1,1
%A _Steven Finch_, Jun 13 2005
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