

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.


1



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


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)


REFERENCES

F. Arndt, Beitrage zur Theorie der quadritischen Formen, Archiv der Mathematik und Physik 15 (1850) 467478.


LINKS

Table of n, a(n) for n=1..54.
A. Cayley, Note sur l'équation x^2  D*y^2 = +4, D=5 (mod. 8), J. Reine Angew. Math. 53 (1857) 369371.
S. 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) 323345.
Wolfdieter Lang, Periods of Indefinite Binary Quadratic Forms, Continued Fractions and the Pell +/4 Equations.


CROSSREFS

Sequence in context: A251537 A004770 A191155 * A107997 A166095 A166090
Adjacent sequences: A107993 A107994 A107995 * A107997 A107998 A107999


KEYWORD

nonn


AUTHOR

Steven Finch, Jun 13 2005


STATUS

approved



