A254930 - OEIS

%I #11 Oct 28 2019 07:44:54

%S 5,7,11,9,13,17,13,19,23,17,15,21,25,17,23,27,35,23,29,21,41,25,31,23,

%T 35,29,39,43,37,31,27,49,53,33,31,37,47,41,55,59,31,45,39,49,37,35,61,

%U 37,35

%N 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).

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

%C 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).

%C 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.

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

%C For comments and the Nagell reference see A254760.

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

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

%e n = 3: 11^2 - 2*7^2 = 23 = A001132(3) = A007522(2).

%e The first pairs of these second class solutions [x2(n), y2(n)] are (a star indicates primes congruent to 1 (mod 8)):

%e n  A001132(n)   a(n)  A254931(n)

%e 1     7           5        3

%e 2    17 *         7        4

%e 3    23          11        7

%e 4    31           9        5

%e 5    41 *        13        8

%e 6    47          17       11

%e 7    71          13        7

%e 8    73 *        19       12

%e 9    89 *        17       10

%e 10   97 *        15        8

%e 11  103          21       13

%e 12  113 *        25       16

%e 13  127          17        9

%e 14  137 *        23       14

%e 15  151          27       17

%e 16  167          35       23

%e 17  191          23       13

%e 18  193 *        29       18

%e 19  199          21       11

%e 20  223          41       27

%e ...

%t 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 *)

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

%K nonn,easy

%O 1,1

%A _Wolfdieter Lang_, Feb 12 2015