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 A118894 Numbers m such that the Pell equation x^2-m*y^2=1 has fundamental solution with x even. 1
 3, 7, 11, 15, 19, 23, 27, 31, 35, 43, 47, 51, 59, 63, 67, 71, 75, 79, 83, 87, 91, 99, 103, 107, 115, 119, 123, 127, 131, 135, 139, 143, 151, 159, 163, 167, 171, 175, 179, 187, 191, 195, 199, 211, 215, 219, 223, 227, 231, 235, 239, 243, 247, 251, 255, 263, 267 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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,... LINKS Wolfdieter Lang, Chebyshev Polynomials and Certain Quadratic Diophantine Equations H. W. Lenstra Jr., Solving the Pell equation, Notices AMS, 49 (2002), 182-192. CROSSREFS Cf. A001075, A001091, A023038, A001081, A001085, A077424, A097310 (x solutions for m=3, 15, 35, 63, 99, 143, 195). Sequence in context: A004767 A131098 A334228 * A194397 A330213 A039957 Adjacent sequences:  A118891 A118892 A118893 * A118895 A118896 A118897 KEYWORD nonn AUTHOR T. D. Noe, May 04 2006 STATUS approved

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Last modified April 12 22:36 EDT 2021. Contains 342933 sequences. (Running on oeis4.)