

A255247


Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2  2*y^2 = A001132(n), n>=1 (primes congruent to {1,7} mod 8).


5



5, 9, 7, 13, 11, 9, 21, 13, 11, 19, 25, 17, 15, 29, 21, 19, 15, 31, 23, 37, 17, 35, 27, 41, 25, 33, 23, 21, 29, 37, 49, 23, 21, 41, 47, 39, 29, 37, 25, 23, 57, 35, 43, 33, 49, 55, 27, 59, 65, 33, 51, 43, 31, 29, 41, 49, 69, 55, 53, 29, 43, 59, 51, 41, 37, 35
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OFFSET

1,1


COMMENTS

For the corresponding term y2(n) see A255248(n).
For the positive fundamental proper (sometimes called primitive) solutions x1(n) and y1(n) of the first class of this (generalized) Pell equation see A255235(n) and A255246(n).
The present solutions of this second class are the next to smallest positive ones. Note that for prime 2 only the first class exists.
For the derivation based on the book of Nagell see the comments on A254934 and A254938 for the primes 1 (mod 8) and 7 (mod 8) separately, where also the Nagell reference is given.


LINKS

Table of n, a(n) for n=1..66.


FORMULA

a(n)^2  2*A255248(n)^2 = A001132(n), n >= 1, gives the second smallest positive (proper) solution of this (generalized) Pell equation.
a(n) = (3*A255235(n+1)  4*A255246(n+1)), n >= 1.


EXAMPLE

The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are
the prime A001132(n) is listed as first entry):
[7, [5, 4]], [17, [9, 7]], [23, [7, 6]],
[31, [13, 10]], [41, [11, 9]], [47, [9, 8]],
[71, [21, 16]], [73, [13, 11]], [79, [11, 10],
[89, [19, 15]], [97, [25, 19]], [103, [17, 14]],
[113, [15, 13]], [127, [29, 22]],
[137, [21, 17]], [151, [19, 16]],
[167, [15, 14]], [191, [31, 24]],
[193, [23, 19]], [199, [37, 28]],
[223, [17, 16]], [233, [35, 27]],
[239, [27, 22]], [241, [41, 31]], ...
n = 1: 5^2  2*4^2 = 25  32 = 7 = A001132(1).
a(3) = (3*3  4*4) = 16  9 = 7.


CROSSREFS

Cf. A001132, A255248, A255235, A255246, A254936, A255233, A254930.
Sequence in context: A232190 A160050 A055566 * A153610 A249385 A247747
Adjacent sequences: A255244 A255245 A255246 * A255248 A255249 A255250


KEYWORD

nonn


AUTHOR

Wolfdieter Lang, Feb 19 2015


EXTENSIONS

More terms from Colin Barker, Feb 26 2015


STATUS

approved



