A227039 Positive solutions x/(2^2*3*89) of the Pell equation x^2 - 73*y^2 = -1.

1, 4562499, 20816392562501, 94974749433621124999, 433322104341376699173125001, 1977031234413227532474550849687499, 9020201052947448468355731925898341687501, 41154649263668650710741676972914857601690249999

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 y(n)/5^3 = A227040(n), n>=0.

References

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

Links

Harvey P. Dale, Table of n, a(n) for n = 0..150

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, 4562499}, 10] (* Harvey P. Dale, Mar 17 2019 *)

Crossrefs

Cf. A227040, A049310.

Sequence in context: A183679 A234793 A227040 * A104950 A234806 A152964

Adjacent sequences: A227036 A227037 A227038 * A227040 A227041 A227042

Keyword

nonn,easy

Author

Wolfdieter Lang, Jun 28 2013

Status

approved