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%I #16 Feb 11 2024 11:37:33
%S 1,1000003,1000005000005,1000007000014000007,
%T 1000009000027000030000009,1000011000044000077000055000011,
%U 1000013000065000156000182000091000013,1000015000090000275000450000378000140000015,1000017000119000442000935001122000714000204000017
%N Positive solutions of the Pell equation x^2 - 89*y^2 = -1. Solutions x = 500*a(n).
%C The Pell equation x^2 - 89*y^2 = -1 has only proper solutions, namely x(n) = 500*a(n) and y(n) = 53*A227138(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 (1000002,-1).
%F 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.
%F a(n) = 2*500001*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*500001*x + x^2).
%e n=0: (500*1)^2 - 89*(53*1)^2 = -1. Proper fundamental (positive) solution.
%e 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.
%t LinearRecurrence[{1000002, -1} {1, 1000003}, 9] (* _Hugo Pfoertner_, Feb 11 2024 *)
%Y Cf. A227138 (y/53 solutions), A049310, A227110, A227111.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Jul 02 2013
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