A227138 Positive solutions of the Pell equation x^2 - 89*y^2 = -1. Solutions y = 53*a(n).

1, 1000001, 1000003000001, 1000005000006000001, 1000007000015000010000001, 1000009000028000035000015000001, 1000011000045000084000070000021000001, 1000013000066000165000210000126000028000001, 1000015000091000286000495000462000210000036000001

Offset

0,2

Comments

The Pell equation x^2 - 89*y^2 = -1 has only proper solutions, namely x(n) = 500*A227137(n) and y(n) = 53*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..8.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (1000002,-1).

Formula

a(n) = S(n, 2*500001) - S(n-1, 2*500001), n >= 0, with the Chebyshev S-polynomials (see A049310) with S(n, -1) = 0. Here 500001 = 3*166667 is the fundamental x solution of the Pell equation x^2 - 89*y^2 = +1.

a(n) = 2*500001*a(n-1) - a(n-2), n >= 1, with inputs a(-1) = 1 and a(0) = 1.

O.g.f.: (1 - x)/(1 - 2*500001*x + x^2).

Example

n=0: (500*1)^2 - 89*(53*1)^2 = -1. Proper fundamental (positive) solution.
n=1: (500*1000003)^2 - 89*(53*1000001)^2 = -1, where 500*1000003 = 500001500 = 2^2*5^3*1000003 and 53*1000001 = 53000053 = 53*101*9901.

Mathematica

LinearRecurrence[{1000002, -1}, {1, 1000001}, 9] (* Hugo Pfoertner, Feb 11 2024 *)

Crossrefs

Cf. A227137 (x/500 solutions), A049310, A227110, A227111.

Sequence in context: A017274 A017514 A017646 * A227111 A227137 A096212

Adjacent sequences: A227135 A227136 A227137 * A227139 A227140 A227141

Keyword

nonn,easy

Author

Wolfdieter Lang, Jul 02 2013

Status

approved