A264439 -id:A264439 - cp's OEIS frontend

A264354 Even nonsquare D values which admit proper solutions to the Pell equation x^2 - D*y^2 = +8.

8, 28, 56, 92, 124, 136, 184, 188, 248, 284, 316, 376, 392, 412, 476, 508, 568, 604, 632, 668, 764, 776, 796, 824, 892, 952, 956, 1016, 1052, 1084, 1148, 1208, 1244, 1288, 1336, 1372, 1436, 1468, 1528, 1532

Offset: 1

Wolfdieter Lang, Nov 18 2015

nonn

This is 4*A261246.
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.
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.

			The first positive proper fundamental solutions of the first class are, when written as [D(n), (x1(n), y1(n))]:
[8, (4, 1)], [28, (6, 1)], [56, (8, 1)], [92, (10, 1)], [124, (78, 7)], [136, (12, 1)], [184, (312, 23)], ...
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):
[8, (4, 1)], [28, (90, 17)], [56, (8, 1)], [92, (470, 49)], [124, (237042, 21287)], [136, (12, 1)], [184, (312, 23)], ...

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

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

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

A264438 One-half of the x member of the positive proper fundamental solution (x = x2(n), y = y2(n)) of the second class for the Pell equation x^2 - D(n)*y^2 = +8 for even D(n) = A264354(n).

Original entry on oeis.org

2, 45, 4, 235, 118521, 6, 156, 665, 8, 410581, 1431, 1464, 10, 217061235, 2629, 20578212225, 12, 143681684300109, 88, 4355, 53946009001, 14, 4149148875801021, 244, 6705, 108, 30839304871, 16, 103789115, 78990793279586649, 9775, 2068, 138751721731, 18, 7987764, 2984191388685, 13661, 5246209297401255, 406200, 5142295

Offset: 1

Wolfdieter Lang, Nov 19 2015

nonn

The corresponding y2(n) value is given by A264439(n). The positive fundamental solution (x1(n), y1(n)) of the first class is given by (2*A261247(n), A261248(n)).
There is only one class of proper solutions for those D = D(n) = A264354(n) that lead to (x1(n), y1(n)) = (x2(n), y2(n)).
See A264354 for comments and examples.

			n=2: D(2) = 28, (2*45)^2 - 28*17^2 = +8. The first class solution was (2*3)^2 - 28*1^2 = +8. This is a D case with two classes of proper solutions.
n=3: D(3) = 56, (2*4)^2 - 56*1^2 = +8. The first class has the same solution, therefore this D has only one class of proper solutions.

Cf. A264354, A261247, A261248, A264439, A263012 (odd D), A264349, A264350, A264351, A264353.

cp's OEIS Frontend

A264354 Even nonsquare D values which admit proper solutions to the Pell equation x^2 - D*y^2 = +8.

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