A227275 Positive solutions of the Pell equation x^2 - 97*y^2 = -1. Solutions y = 569*a(n).

1, 125619265, 15780199864759489, 1982297124344386149095809, 249014709754052503489782225496705, 31281045062507114031551757322356418422721, 3929501920465067535407122841792035182868864246081, 493621146994412131157226720050635552766405178489920173825

Offset

0,2

Comments

The Pell equation x^2 - 97*y^2 = -1 has only proper solutions, namely x(n) = 5604*A227274(n) and y(n) = 569*a(n), n >= 0.

References

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, ch. VI, 57., pp. 201-204.

O. Perron, Die Lehre von den Kettenbruechen, Band I, Teubner, Stuttgart, 1954, Paragraph 27, pp. 92-95.

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 (125619266,-1).

Formula

a(n) = S(n, 2*62809633) - S(n-1, 2*62809633), n >= 0, with the Chebyshev S-polynomials (see A049310) with S(n, -1) = 0. Here 62809633, a prime, is the fundamental x solution of the Pell equation x^2 - 97*y^2 = +1.

a(n) = 2*62809633*a(n-1) - a(n-2), n >= 1, with inputs a(-1) = 1 and a(0) = 1.

O.g.f.: (1 - x)/(1 - 2*62809633*x + x^2).

Example

n=0: (5604*1)^2 - 97*(569*1)^2 = -1. Proper fundamental (positive) solution.
n=1: (5604*125619267)^2 - 97*(569*125619265)^2 = -1, where 5604*125619267 = (2^2*3*467)*(3*41873089) = 703970372268 and 569*125619265  = 569*(5*401*62653) = 71477361785.

Mathematica

LinearRecurrence[{125619266, -1}, {1, 125619265}, 20] (* Harvey P. Dale, Aug 27 2017 *)

Crossrefs

Cf. A227274 (x/5604 solutions), A049310, A227150, A227151.

Sequence in context: A269119 A068249 A289552 * A227151 A227274 A162450

Adjacent sequences: A227272 A227273 A227274 * A227276 A227277 A227278

Keyword

nonn,easy

Author

Wolfdieter Lang, Jul 05 2013

Status

approved