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A227110 Nonnegative solutions of the Pell equation x^2 - 89*y^2 = +1. Solutions x = a(n). 3
1, 500001, 500002000001, 500003000004500001, 500004000010000008000001, 500005000017500025000012500001, 500006000027000056000052500018000001, 500007000038500105000147000098000024500001, 500008000052000176000330000336000168000032000001 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The Pell equation x^2 - 89*y^2 = +1 has only proper solutions, namely x(n) = a(n) and y(n) = 53000*A227111(n), n>= 0.

REFERENCES

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, ch. VI, 56., pp. 115-200.

O. Perron, Die Lehre von den Kettenbruechen, Band I, Teubner, Stuttgart, 1954, Paragraph 27, p. 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-2, 2*500001))/2 = T(n, 500001) with the Chebyshev S- and T-polynomials (see A049310 and A053120, respectively). S(n, -2) = -1, 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 input a(-1) = 500001 and a(0) = 1.

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

EXAMPLE

n=0: 1^2 - 89*0^2  = +1 (proper, but not a positive solution),

n=1: 500001^2 - 89*53000^2 = +1, where  53000 = 2^3*5^3*53 is the positive fundamental y-solution and 500001 = 3*166667 the corresponding fundamental x-solution.

n=2: 500002000001^2 - 89*53000106000^2 = +1, where 500002000001 = 7^2*17*600242497 and 53000106000 = 53000*1000002 = (2^3*5^3*53)*(2*3*166667).

CROSSREFS

Cf.:  A227111 (y/53000 solutions), A049310, A053120, A227137, A227138.

Sequence in context: A238152 A145539 A157759 * A187313 A183789 A203548

Adjacent sequences:  A227107 A227108 A227109 * A227111 A227112 A227113

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jul 02 2013

STATUS

approved

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Last modified November 30 03:03 EST 2021. Contains 349416 sequences. (Running on oeis4.)