A107996 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.

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

Offset: 1

Author

Steven Finch, Jun 13 2005

Keywords

• nonn

Comments

From Wolfdieter Lang, Oct 30 2015: (Start)

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)

Links

• F. Arndt, Beiträge zur Theorie der quadratischen Formen, Archiv der Mathematik und Physik 15 (1850) 467-478.
• A. Cayley, Note sur l'équation x^2 - D*y^2 = +-4, D=5 (mod. 8), J. Reine Angew. Math. 53 (1857) 369-371.
• Steven R. Finch, Class number theory
• Steven R. Finch, Class number theory [Cached copy, with permission of the author]
• N. Ishii, P. Kaplan and K. S. Williams, On Eisenstein's problem, Acta Arith. 54 (1990) 323-345.
• Wolfdieter Lang, Periods of Indefinite Binary Quadratic Forms, Continued Fractions and the Pell +/-4 Equations.

Crossrefs

Cf. A079896.