A255233 - OEIS

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).

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

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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.

LINKS

Table of n, a(n) for n=1..67.

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]],