A118894 Numbers m such that the Pell equation x^2-m*y^2=1 has fundamental solution with x even.

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

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

Table of n, a(n) for n=1..57.

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