A254931 Fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = A001132(n), n >= 1, (primes congruent to 1 or 7 mod 8).

3, 4, 7, 5, 8, 11, 7, 12, 15, 10, 8, 13, 16, 9, 14, 17, 23, 13, 18, 11, 27, 14, 19, 12, 22, 17, 25, 28, 23, 18, 14, 32, 35, 19, 17, 22, 30, 25, 36, 39, 16, 28, 23, 31, 21, 19, 40, 20, 18, 38

Offset: 1

Author

Wolfdieter Lang, Feb 12 2015

Keywords

• nonn
• easy

Comments

The corresponding terms x = x2(n) are given in A254930(n).

The y2-sequence for the second class for the primes congruent to 1 (mod 8), which are given in A007519, is 2*A254763. For the primes congruent to 7 (mod 8), given in A007522, the y2-sequence is A254929.

For comments and the Nagell reference see A254760.

Examples

			a(4) = 2*7 - 3*3 = 5.
A254930(4)^2 - 2*a(4)^2 = 9^2 - 2*5^2 = 31 = A001132(4) = A007522(3).
See A254930 for the first pairs (x2(n), y2(n)).
		

Crossrefs

Cf. A254930, A001132, A002334, A002335, A007519, 2*A254763, A007522, A254929, A254760.

Programs

Mathematica

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[MatchQ[Mod[p, 8], 1|7], rp = Reduce[x > 0 && y > 0 && x^2 - 2 y^2 == p, {x, y}, Integers]; If[rp =!= False, xy = {x, y} /. {ToRules[rp /. C[1] -> 1]}; y2 = xy[[-1, 2]] // Simplify; Print[y2]; Sow[y2]]]]][[2, 1]] (* Jean-François Alcover, Oct 28 2019 *)

Formula

A254930(n)^2 - 2*a(n)^2 = A001132(n), and a(n) is the second largest (proper) positive integer satisfying this (generalized) Pell equation.

a(n) = 2*A002334(n+1) - 3*A002335(n+1), n >= 1.