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A254764 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).

%I A254764 #18 Feb 19 2015 12:14:57
%S A254764 3,5,7,7,11,9,11,15,13,13,17,19,15,17,19,17,19,19,23,25,23,21,25,23,
%T A254764 27,29,29,25,27,35,31,27,29,33,29,29,31,35,31,37,43,35,33,37,33,35,33,
%U A254764 41,47,35
%N A254764 Fundamental positive solution x = x1(n) of the first class of the Pell equation x^2 - 2*y^2 = A007522(n), n >=1 (primes congruent to 7 mod 8).
%C A254764 For the corresponding term y1(n) see A254765(n).
%C A254764 For the positive fundamental proper (sometimes called primitive) solutions x2(n) and y2(n) of the second class of this (generalized) Pell equation see A254766(n) and A254929(n).
%C A254764 The present solutions of the first class are the smallest positive ones.
%C A254764 See the Nagell reference Theorem 111 p. 210 for the proof of the existence of solutions (the discriminant of this binary quadratic form is +8 hence it is an indefinite form with an infinitude of solutions if there exists at least one).
%C A254764 See the Nagell reference Theorem 110, p. 208 for the proof that there are only two classes of solutions for this Pell equation, because the equation is solvable, and the primes A007522(n) do not divide 4.
%C A254764 The present fundamental solutions are found according to the Nagell reference Theorem 108, p. 205, adapted to the case at hand, by scanning the following two inequalities for solutions x1(n) = 2*X1(n) + 1 and y1(n) = 2*Y1(n) + 1. The intervals for X1(n) and Y1(n) to be scanned are ceiling((sqrt(2+p(n))-1)/2) <= X1(n) <=  floor(sqrt((2*p(n))-1)/2), with p(n) = A007522(n) and 0 <= Y1(n) <= floor((sqrt(p(n)/2)-1)/2).
%C A254764 The general positive proper solutions for both classes are obtained by applying positive powers of the matrix M = [[3,4],[2,3]] on the fundamental column vectors (x(n),y(m))^T.
%C A254764 The least positive x solutions (that is the ones of the first class) for the primes +1 and -1 (mod 8) together (including also prime 2) are given in A002334.
%D A254764 T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.
%F A254764 a(n)^2 - 2*A254765(n)^2 = A007522(n) gives the smallest positive (proper) solution of this (generalized) Pell equation.
%e A254764 The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (we list the prime A007522(n) as first entry):
%e A254764   [7, [3, 1]], [23, [5, 1]], [31, [7, 3]], [47, [7, 1]], [71, [11, 5]], [79, [9, 1]], [103, [11, 3]], [127, [15, 7]], [151, [13, 3]], [167, [13, 1]], [191, [17, 7]], [199, [19, 9]], [223, [15, 1]], ...
%e A254764 a(3)^2 - 2*A254765(3)^2 = 7^2 - 2*3^2 = 31 = A007522(3).
%Y A254764 Cf. A007522, A254765, A254766, A254929, A254760, A254761, A254762, A254763, A002334.
%K A254764 nonn,easy
%O A254764 1,1
%A A254764 _Wolfdieter Lang_, Feb 12 2015