A227040 Positive solutions y/5^3 of the Pell equation x^2 - 73*y^2 = -1.

1, 4562497, 20816383437505, 94974707800845124993, 433321914391919464706875009, 1977030367769208799178386969687489, 9020197098885846285919400272960522312513, 41154631223270498877446922697782658742826249985

Offset

0,2

Comments

The proper positive solutions of the Pell equation x^2 - 73*y^2 = -1 start with the fundamental solution (x_0, y_0) = (1068, 125). 1068 = 2^2*3*89, 125 = 5^3. The solutions x(n)/1068 = A227039(n), n>=0.

References

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, ch. Vi, 58., p. 204-212.

Links

Table of n, a(n) for n=0..7.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (4562498,-1).

Formula

a(n) = S(n,4562498) - S(n-1,4562498), n >= 0, with the Chebyshev S-polynomials (A049310), with S(-1,x) = 0. 4562498 = 2*2281249 is the fundamental (improper) u solution of u^2 - 73*v^3 = +4 (together with the positive v = 53400 = 2*26700).

O.g.f.: (1 - x)/(1 - 4562498*x + x^2).

a(n) = 4562498*a(n-1) - a(n-2), n >= 1, a(-1) = 1, a(0) = 1.

Example

n=0: (2^2*3*89*1)^2 - 73*(5^3*1)^2 = -1.
n=1: (2^2*3*89*4562499)^2 - 73*(5^3*4562497)^2 = -1. 4562499 = 3*67*22699, 4562497 is prime.

Mathematica

LinearRecurrence[{4562498, -1}, {1, 4562497}, 20] (* Harvey P. Dale, Oct 08 2017 *)

Crossrefs

Cf. A227039, A049310.

Sequence in context: A344831 A183679 A234793 * A227039 A104950 A234806

Adjacent sequences: A227037 A227038 A227039 * A227041 A227042 A227043

Keyword

nonn,easy

Author

Wolfdieter Lang, Jun 28 2013

Status

approved