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%I #7 Jan 05 2023 11:06:06
%S 3,7,11,15,19,23,27,31,35,43,47,51,59,63,67,71,75,79,83,87,91,99,103,
%T 107,115,119,123,127,131,135,139,143,151,159,163,167,171,175,179,187,
%U 191,195,199,211,215,219,223,227,231,235,239,243,247,251,255,263,267
%N Numbers m such that the Pell equation x^2-m*y^2=1 has fundamental solution with x even.
%C 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 Wolfdieter Lang, <a href="http://www.itp.kit.edu/~wl/p36pub/p36.pdf">Chebyshev Polynomials and Certain Quadratic Diophantine Equations</a>
%H H. W. Lenstra Jr., <a href="http://www.ams.org/notices/200202/fea-lenstra.pdf">Solving the Pell equation</a>, Notices AMS, 49 (2002), 182-192.
%Y Cf. A001075, A001091, A023038, A001081, A001085, A077424, A097310 (x solutions for m=3, 15, 35, 63, 99, 143, 195).
%K nonn
%O 1,1
%A _T. D. Noe_, May 04 2006
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