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 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). 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 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]}; 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: A096919 A023594 A277777 * A317769 A104200 A249916 Adjacent sequences:  A254927 A254928 A254929 * A254931 A254932 A254933 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Feb 12 2015 STATUS approved

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Last modified November 20 23:06 EST 2019. Contains 329348 sequences. (Running on oeis4.)