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
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
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jul 02 2013
STATUS
approved