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A107996
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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.
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1
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5, 13, 21, 29, 45, 53, 61, 69, 77, 85, 93, 109, 117, 125, 133, 149, 157, 165, 173, 181, 205, 213, 221, 229, 237, 245, 253, 261, 277, 285, 293, 301, 309, 317, 341, 357, 365, 397, 413, 421, 429, 437, 445, 453, 461, 469, 477, 493, 501, 509, 517, 525, 533, 541
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OFFSET
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1,1
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COMMENTS
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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)).
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.
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).
(End)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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