A263011 - OEIS

%I #13 Dec 12 2015 02:11:40

%S 17,41,73,89,97,113,137,161,193,217,233,241,257,281,313,329,337,353,

%T 401,409,433,449,457,497,521,553,569,577,593,601,617,641,673,697,713,

%U 721,761,769,809,833,857,881,889,929,937,953,977,1009,1033,1049,1057,1081,1097,1129,1153,1169,1193,1201,1217

%N Numbers D == 1 (mod 8), not a square, and if composite without prime factors 3 or 5 (mod 8).

%C These numbers are the odd D candidates for the (generalized) Pell equation x^2 - D*y^2 = +8 which could have proper solutions (x, y) with x and y both odd (and gcd(x, y) = 1).

%C Proof: Put x =2*X + 1, y = 2*Y + 1; then 8*(T(X) - D*T(Y)) = 8 - 1 + D = 7 + D, with the triangular numbers T = A000217. Hence, D == -7 (mod 8) == +1 (mod 8). Only nonsquare numbers D are considered for the Pell equation (square D leads to a factorization with only one solution: D = 1, (x, y) = (3, 1)). Consider a prime factor p == 3 or 5 (mod 8) (A007520 or A007521) of D. Then x^2 == 8 (mod p). Because the Legendre symbol (8/p) = (2*2^2/p) = (2/p) ==  (-1)^(p^2-1)/8 (see, e.g., Nagell, eq. (3), p. 138) this becomes -1 for these primes p, and therefore a candidate for D cannot have any prime factors 3 or 5 (mod 8).

%C However, not all of these candidates admit solutions. For the exceptions see A264348.

%C The remaining Ds (that admit solutions) are given in A263012.

%D T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.

%H Michael De Vlieger, <a href="/A263011/b263011.txt">Table of n, a(n) for n = 1..10000</a>

%t Select[8 Range@ 154 + 1, Or[PrimeQ@ #, CompositeQ@ # && AllTrue[Union@ Mod[First /@ FactorInteger@ #, 8], ! MemberQ[{3, 5}, #] &]] && ! IntegerQ@ Sqrt@ # &] (* _Michael De Vlieger_, Dec 11 2015, Version 10 *)

%Y Cf. A263012, A264348.

%K nonn,easy

%O 1,1

%A _Wolfdieter Lang_, Nov 17 2015