

A263010


Exceptional odd numbers D that do not admit a solution to the Pell equation X^2  D Y^2 = +2.


2



791, 799, 943, 1271, 1351, 1631, 1751, 1967, 2159, 2303, 2359, 2567, 3143, 3199, 3503, 3703, 3983, 4063, 4439, 4471, 4559, 4607, 4711, 5047, 5183, 5207, 5359, 5663, 5911, 5983, 6511, 6671, 6839, 7063, 7231, 7663, 7871, 8183, 8407, 8711, 9143, 9271, 9751, 9863, 10183, 10367
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OFFSET

1,1


COMMENTS

These are the odd numbers 7 (mod 8), not a square, that have in the composite case no prime factors 3 or 5 (mod 8), and do not represent +2 by the indefinite binary quadratic form X^2  D*Y^2 (with discriminant 4*D > 0).
The numbers D which admit solutions of the Pell equation X^2  D Y^2 = +2 are given by A261246.
Necessary conditions for nonsquare odd D were shown there to be D == 7 (mod 8), without prime factors 3 or 5 (mod 8) in the composite case. Thus only prime factors +1 (mod 8) and 1 (mod 8) can appear, and the number of the latter is odd. It has been conjectured that all such numbers D appear in A261246, but this conjecture is false as the present sequence shows.
All entries seem to be composite. The first numbers are 791 = 7*113, 799 = 17*47, 943 = 23*41, 1271 = 31*41, 1351 = 7*193, 1631 = 7*233, ...
For counterexamples to the conjecture in A261246 for even D see A264352.


LINKS

Table of n, a(n) for n=1..46.


CROSSREFS

Cf. A261246, A261247, A261248, A264352.
Sequence in context: A182387 A237735 A252517 * A351670 A282547 A104398
Adjacent sequences: A263007 A263008 A263009 * A263011 A263012 A263013


KEYWORD

nonn


AUTHOR

Wolfdieter Lang, Nov 10 2015


STATUS

approved



