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%I #14 Dec 12 2015 00:51:01
%S 791,799,943,1271,1351,1631,1751,1967,2159,2303,2359,2567,3143,3199,
%T 3503,3703,3983,4063,4439,4471,4559,4607,4711,5047,5183,5207,5359,
%U 5663,5911,5983,6511,6671,6839,7063,7231,7663,7871,8183,8407,8711,9143,9271,9751,9863,10183,10367
%N Exceptional odd numbers D that do not admit a solution to the Pell equation X^2 - D Y^2 = +2.
%C 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).
%C The numbers D which admit solutions of the Pell equation X^2 - D Y^2 = +2 are given by A261246.
%C 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.
%C 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, ...
%C For counterexamples to the conjecture in A261246 for even D see A264352.
%Y Cf. A261246, A261247, A261248, A264352.
%K nonn
%O 1,1
%A _Wolfdieter Lang_, Nov 10 2015
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