A254766 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, 11, 9, 17, 13, 23, 21, 17, 27, 35, 23, 21, 41, 31, 29, 39, 37, 53, 33, 31, 41, 59, 39, 49, 37, 35, 43, 63, 53, 37, 49, 77, 59, 47, 75, 83, 65, 53, 73, 51, 45, 61, 71, 59, 79, 69, 95, 55, 49, 101

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) = 7 + 8*A139487(n) is given in A254929(n).

The positive fundamental solutions of the first classes are given in (A254764(n), A254765(n)).

For comments and the Nagell reference see A254764.

Links

Table of n, a(n) for n=1..50.

Formula

a(n)^2 - 2*(A254929(n))^2 = A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = 3*A254764(n) - 4*A254765(n), n >= 1.

Example

The first pairs [x2(n), y2(n)] of the fundamental positive solutions of the second class are (we list the prime A007522(n) as first entry): [7,[5,3]], [23,[11,7]], [31,[9,5]], [47,[17,11]], [71,[13,7]], [79,[23,15]], [103,[21,13]], [127,[17,9]], [151,[27,17]], [167,[35,23]], [191,[23,13]], [199,[21,11]], [223,[41,27]], [239,[31,19]], [263,[29,17]], [271,[39,25]], ...

Crossrefs

Cf. A007522, A139487, A254929, A254764, A254765, A254760, A254761, A254762, A254763,

Sequence in context: A113964 A356480 A075261 * A185201 A112956 A258995

Adjacent sequences: A254763 A254764 A254765 * A254767 A254768 A254769

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 11 2015

Status

approved