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A264354 Even nonsquare D values which admit proper solutions to the Pell equation x^2 - D*y^2 = +8. This is 4*A261246. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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 A216246.

REFERENCES

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

LINKS

Table of n, a(n) for n=1..40.

FORMULA

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

EXAMPLE

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)], ...

CROSSREFS

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

Sequence in context: A229393 A173681 A045850 * A033580 A299289 A212515

Adjacent sequences:  A264351 A264352 A264353 * A264355 A264356 A264357

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Nov 18 2015

STATUS

approved

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Last modified July 20 05:56 EDT 2019. Contains 325168 sequences. (Running on oeis4.)