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A255233
Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).
6
5, 7, 13, 9, 21, 11, 17, 29, 19, 15, 31, 37, 17, 27, 33, 23, 29, 21, 41, 47, 37, 23, 43, 33, 49, 55, 51, 31, 41, 69, 53, 29, 43, 59, 35, 31, 45, 61, 41, 67, 85, 57, 47, 63, 43, 53, 35, 75, 93, 37, 71, 61, 83, 47, 89, 39, 73, 53, 63, 79, 49, 85, 69, 97, 103, 109, 55
OFFSET
1,1
COMMENTS
The corresponding term y = y2(n) of this fundamental solution of the second class of the (generalized) Pell equation x^2 - 2*y^2 = -A007522(n) = -(1 + A139487(n)*8) is given in 2*A255234(n).
For comments and the Nagell reference see A254938.
FORMULA
a(n)^2 - 2*(2*A255234(n))^2 = -A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.
a(n) = -(3*A254938(n) - 4*2*A255232(n)), n >= 1.
EXAMPLE
The first pairs [x2(n), y2(n)] of the fundamental positive solutions of this second class are (the prime A007522(n) appears as first entry):
[7, [5, 4]], [23, [7, 6]], [31, [13, 10]],
[47, [9, 8]], [71, [21, 16]], [79, [11, 10]], [103, [17, 14]], [127, [29, 22]],
[151, [19, 16]], [167, [15, 14]],
[191, [31, 24]], [199, [37, 28]],
[223, [17, 16]], [239, [27, 22]],
[263, [33, 26]], [271, [23, 20]],
[311, [29, 24]], [359, [21, 20]],
[367, [41, 32]], [383, [47, 36]],
[431, [37, 30]], [439, [23, 22]],
[463, [43, 34]], [479, [33, 28]], ...
n= 4: 9^2 - 2*(2*4)^2 = -47 = -A007522(4).
a(4) = -(3*5 - 4*(2*3)) = 24 - 15 = 9.
CROSSREFS
KEYWORD
nonn,look,easy
AUTHOR
Wolfdieter Lang, Feb 18 2015
EXTENSIONS
More terms from Colin Barker, Feb 23 2015
STATUS
approved