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 A255247 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 {1,7} mod 8). 5

%I

%S 5,9,7,13,11,9,21,13,11,19,25,17,15,29,21,19,15,31,23,37,17,35,27,41,

%T 25,33,23,21,29,37,49,23,21,41,47,39,29,37,25,23,57,35,43,33,49,55,27,

%U 59,65,33,51,43,31,29,41,49,69,55,53,29,43,59,51,41,37,35

%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 {1,7} mod 8).

%C For the corresponding term y2(n) see A255248(n).

%C For the positive fundamental proper (sometimes called primitive) solutions x1(n) and y1(n) of the first class of this (generalized) Pell equation see A255235(n) and A255246(n).

%C The present solutions of this second class are the next to smallest positive ones. Note that for prime 2 only the first class exists.

%C For the derivation based on the book of Nagell see the comments on A254934 and A254938 for the primes 1 (mod 8) and 7 (mod 8) separately, where also the Nagell reference is given.

%F a(n)^2 - 2*A255248(n)^2 = -A001132(n), n >= 1, gives the second smallest positive (proper) solution of this (generalized) Pell equation.

%F a(n) = -(3*A255235(n+1) - 4*A255246(n+1)), n >= 1.

%e The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are

%e the prime A001132(n) is listed as first entry):

%e [7, [5, 4]], [17, [9, 7]], [23, [7, 6]],

%e [31, [13, 10]], [41, [11, 9]], [47, [9, 8]],

%e [71, [21, 16]], [73, [13, 11]], [79, [11, 10],

%e [89, [19, 15]], [97, [25, 19]], [103, [17, 14]],

%e [113, [15, 13]], [127, [29, 22]],

%e [137, [21, 17]], [151, [19, 16]],

%e [167, [15, 14]], [191, [31, 24]],

%e [193, [23, 19]], [199, [37, 28]],

%e [223, [17, 16]], [233, [35, 27]],

%e [239, [27, 22]], [241, [41, 31]], ...

%e n = 1: 5^2 - 2*4^2 = 25 - 32 = -7 = -A001132(1).

%e a(3) = -(3*3 - 4*4) = 16 - 9 = 7.

%Y Cf. A001132, A255248, A255235, A255246, A254936, A255233, A254930.

%K nonn

%O 1,1

%A _Wolfdieter Lang_, Feb 19 2015

%E More terms from _Colin Barker_, Feb 26 2015

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Last modified November 16 16:51 EST 2019. Contains 329200 sequences. (Running on oeis4.)