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Positive solutions of the Pell equation x^2 - 97*y^2 = -1. Solutions y = 569*a(n).
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%I #13 Feb 12 2024 06:00:53

%S 1,125619265,15780199864759489,1982297124344386149095809,

%T 249014709754052503489782225496705,

%U 31281045062507114031551757322356418422721,3929501920465067535407122841792035182868864246081,493621146994412131157226720050635552766405178489920173825

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

%C 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.

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

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

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (125619266,-1).

%F 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.

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

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

%e n=0: (5604*1)^2 - 97*(569*1)^2 = -1. Proper fundamental (positive) solution.

%e 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.

%t LinearRecurrence[{125619266,-1},{1,125619265},20] (* _Harvey P. Dale_, Aug 27 2017 *)

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

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jul 05 2013