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A227137
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Positive solutions of the Pell equation x^2 - 89*y^2 = -1. Solutions x = 500*a(n).
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3
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1, 1000003, 1000005000005, 1000007000014000007, 1000009000027000030000009, 1000011000044000077000055000011, 1000013000065000156000182000091000013, 1000015000090000275000450000378000140000015, 1000017000119000442000935001122000714000204000017
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OFFSET
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0,2
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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MATHEMATICA
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LinearRecurrence[{1000002, -1} {1, 1000003}, 9] (* Hugo Pfoertner, Feb 11 2024 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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