A254930 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 or 7 mod 8).

5, 7, 11, 9, 13, 17, 13, 19, 23, 17, 15, 21, 25, 17, 23, 27, 35, 23, 29, 21, 41, 25, 31, 23, 35, 29, 39, 43, 37, 31, 27, 49, 53, 33, 31, 37, 47, 41, 55, 59, 31, 45, 39, 49, 37, 35, 61, 37, 35

Offset

1,1

Comments

The corresponding terms y = y2(n) are given in A254931(n).

There is only one fundamental solution for prime 2 (no second class exists), and this solution (x, y) has been included in (A002334(1), A002335(1)) = (2, 1).

The second class x sequence for the primes 1 (mod 8), which are given in A007519, is A254762, and for the primes 7 (mod 8), given in A007522, it is A254766.

The second class solutions give the second smallest positive integer solutions of this Pell equation.

For comments and the Nagell reference see A254760.

Links

Table of n, a(n) for n=1..49.

Formula

a(n)^2 - 2*A254931(n)^2 = A001132(n), and a(n) is the second largest (proper) positive integer solving this (generalized) Pell equation.

a(n) = 3*A002334(n+1) - 4*A002335(n+1), n >= 1.

Example

n = 3: 11^2 - 2*7^2 = 23 = A001132(3) = A007522(2).
The first pairs of these second class solutions [x2(n), y2(n)] are (a star indicates primes congruent to 1 (mod 8)):
n  A001132(n)   a(n)  A254931(n)
1     7           5        3
2    17 *         7        4
3    23          11        7
4    31           9        5
5    41 *        13        8
6    47          17       11
7    71          13        7
8    73 *        19       12
9    89 *        17       10
10   97 *        15        8
11  103          21       13
12  113 *        25       16
13  127          17        9
14  137 *        23       14
15  151          27       17
16  167          35       23
17  191          23       13
18  193 *        29       18
19  199          21       11
20  223          41       27
...

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]}; x2 = xy[[-1, 1]] // Simplify; Print[x2]; Sow[x2]]]]][[2, 1]] (* Jean-François Alcover, Oct 28 2019 *)

Crossrefs

Cf. A001132, A254931, A002334, A002335, A007519, A254762, A007522, A254766, A254760.

Sequence in context: A023594 A357099 A277777 * A317769 A104200 A249916

Adjacent sequences: A254927 A254928 A254929 * A254931 A254932 A254933

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 12 2015

Status

approved