%I A118894
%S A118894 3,7,11,15,19,23,27,31,35,43,47,51,59,63,67,71,75,79,83,87,91,99,103,
%T A118894 107,115,119,123,127,131,135,139,143,151,159,163,167,171,175,179,187,
%U A118894 191,195,199,211,215,219,223,227,231,235,239,243,247,251,255,263,267
%N A118894 Numbers m such that the Pell equation x^2-m*y^2=1 has fundamental solution 
               with x even.
%C A118894 Numbers m such that A002350(m) is even. These m can be used to generate 
               consecutive odd powerful numbers, as in A076445. As shown by Lang, 
               the solution of Pell's equation is greatly simplified by Chebyshev 
               polynomials of the first kind T(n,x), which is illustrated in A001075 
               for the case m=3. In that case, the solutions are x=T(n,2), for integer 
               n>0. For any m in this sequence, let E(k)=T(m+2mk,A002350(m)). Then 
               E(k)-1 and E(k)+1 are consecutive odd powerful numbers for k=0,1,
               2,...
%H A118894 Wolfdieter Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/
               p36pub/p36.pdf">Chebyshev Polynomials and Certain Quadratic Diophantine 
               Equations</a>
%H A118894 H. W. Lenstra Jr., <a href="http://www.ams.org/notices/200202/fea-lenstra.pdf">
               Solving the Pell equation, Notices AMS, 49 (2002), 182-192.</a>
%Y A118894 Cf. A001075, A001091, A023038, A001081, A001085, A077424, A097310 (x 
               solutions for m=3, 15, 35, 63, 99, 143, 195).
%Y A118894 Sequence in context: A103543 A004767 A131098 this_sequence A039957 A079422 
               A022797
%Y A118894 Adjacent sequences: A118891 A118892 A118893 this_sequence A118895 A118896 
               A118897
%K A118894 nonn
%O A118894 1,1
%A A118894 T. D. Noe (noe(AT)sspectra.com), May 04 2006