OFFSET
0,3
COMMENTS
The Pell equation x^2 - 97*y^2 = +1 has only proper solutions, namely x(n) = A227150(n) and y(n) = 6377352*a(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, pp. 92-95.
LINKS
FORMULA
a(n) = S(n-1,2*62809633), n >= 0, with the Chebyshev S-polynomials (see A049310). S(n,-1) = 0.
a(n) = 2*62809633*a(n-1) - a(n-2), n >= 1, with input a(-1) = -1 and a(0) = 0.
O.g.f.: x/(1 - 2*62809633*x + x^2).
EXAMPLE
n=0: 1^2 - 97*0^2 = +1 (a proper, but not a positive solution),
n=1: 62809633^2 - 97*6377352^2 = +1, where 62809633 is prime and 6377352 = 2^3*3*467*569 is the positive fundamental y-solution.
n=2: 7890099995189377^2 - 97*801118277263632^2 = +1, where 801118277263632 = 6377352*125619266 = (2^3*3*467*569)*(2*62809633).
MATHEMATICA
LinearRecurrence[{125619266, -1}, {0, 1}, 20] (* Harvey P. Dale, Oct 15 2014 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jul 05 2013
STATUS
approved