A264354 - OEIS

%I #20 Jul 27 2024 02:54:28

%S 8,28,56,92,124,136,184,188,248,284,316,376,392,412,476,508,568,604,

%T 632,668,764,776,796,824,892,952,956,1016,1052,1084,1148,1208,1244,

%U 1288,1336,1372,1436,1468,1528,1532

%N Even nonsquare D values which admit proper solutions to the Pell equation x^2 - D*y^2 = +8.

%C This is 4*A261246.

%C The proper positive fundamental solution (x1(n), y1(n)) of the first class is given by (2*A261247(n), A261248(n)) for D(n) = a(n), n >= 1. If there are two classes the proper positive fundamental solution (x2(n), y2(n)) for the second class is given by (A264357(n), A264386(n)) for D(n). If the fundamental solutions of the two classes coincide then there is only one class (the ambiguous case) for these D(n) values. It is conjectured that there are no more than two classes. For the computation of (x2(n), y2(n)) from (x1(n), -y1(n)) by application of the matrix M(n) for D(n) see a comment under A263012.

%C D = 8, 56, 136, 184, 248, 376, 392, 568, 632, 776, 824, 952, 1016, 1208, 1288, 1336, 1528, ... have only one class of solution, because for them (x1, y1) = (x2, y2). These D values are the ones with x1(n) = 2*sqrt(x0(n)+1) and y1(n) = 2*y0(n) / sqrt(x0(n)+1) where (x0(n), y0(n)) are the positive fundamental solution of the +1 Pell equation with D = D(n). These are the upper bounds of the inequalities, eqs. (4) and (5) given in the Nagell reference on p. 206. E.g., D = 184 = A000037(171) = a(8) with x0(8) = A033313(171) = 24335 and y0(8) = A033317(171) = 1794 leads to x1(8) = 2*sqrt(24336) = 312 and y1(8) = 2*1794/sqrt(24336) = 23. These D numbers with only one class of proper solutions are the entries which are divisible by 8, that is four times the even numbers of A261246.

%D T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New Tork, 1964, p. 206.

%F a(n) = 4*A261246(n).

%e The first positive proper fundamental solutions of the first class are, when written as [D(n), (x1(n), y1(n))]:

%e [8, (4, 1)], [28, (6, 1)], [56, (8, 1)], [92, (10, 1)], [124, (78, 7)], [136, (12, 1)], [184, (312, 23)], ...

%e The first positive proper fundamental solutions of the second class [D(n), (x2(n), y2(n))] are (if the values for both classes coincide there is only one class):

%e [8, (4, 1)], [28, (90, 17)], [56, (8, 1)], [92, (470, 49)], [124, (237042, 21287)], [136, (12, 1)], [184, (312, 23)], ...

%Y Cf. A000037, A033313, A033317, A261246, A263012 (odd D), A261247 (x1/2), A261248 (y1), A264438 (x2), A264439 (y2), A264355.

%K nonn

%O 1,1

%A _Wolfdieter Lang_, Nov 18 2015