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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
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
KEYWORD
nonn
AUTHOR
T. D. Noe, May 04 2006
STATUS
approved

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Last modified March 28 10:31 EDT 2024. Contains 371240 sequences. (Running on oeis4.)