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A227274
Positive solutions of the Pell equation x^2 - 97*y^2 = -1. Solutions x = 5604*a(n).
2
1, 125619267, 15780200115998021, 1982297155904786129853319, 249014713718646783738954504445833, 31281045560536537504251044551093148365259, 3929501983027158158450774377594837056318431034061, 493621154853416034649452413908532772153277417677215453967
OFFSET
0,2
COMMENTS
The Pell equation x^2 - 97*y^2 = -1 has only proper solutions, namely x(n) = 5604*a(n) and y(n) = 569*A227275(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.
FORMULA
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.
a(n) = 2*62809633*a(n-1) - a(n-2), n >= 1, with inputs a(-1) = -1 and a(0) = 1.
O.g.f.: (1 + x)/(1 - 2*62809633*x + x^2).
EXAMPLE
n=0: (5604*1)^2 - 97*(569*1)^2 = -1. Proper fundamental (positive) solution.
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.
MATHEMATICA
LinearRecurrence[{125619266, -1}, {1, 125619267}, 8] (* Hugo Pfoertner, Feb 11 2024 *)
CROSSREFS
Cf. A227275 (y/569 solutions), A049310, A227150, A227151.
Sequence in context: A289552 A227275 A227151 * A162450 A339537 A319065
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jul 05 2013
STATUS
approved