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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).

%I A255247 #25 May 18 2025 04:08:08
%S A255247 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 A255247 25,33,23,21,29,37,49,23,21,41,47,39,29,37,25,23,57,35,43,33,49,55,27,
%U A255247 59,65,33,51,43,31,29,41,49,69,55,53,29,43,59,51,41,37,35
%N 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).
%C A255247 For the corresponding term y2(n) see A255248(n).
%C A255247 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 A255247 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 A255247 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 A255247 a(n)^2 - 2*A255248(n)^2 = -A001132(n), n >= 1, gives the second smallest positive (proper) solution of this (generalized) Pell equation.
%F A255247 a(n) = -(3*A255235(n+1) - 4*A255246(n+1)), n >= 1.
%e A255247 The first pairs [x1(n), y1(n)] of the fundamental positive solutions of this first class are (the prime A001132(n) is listed as first entry):
%e A255247   [7, [5, 4]], [17, [9, 7]], [23, [7, 6]],
%e A255247   [31, [13, 10]], [41, [11, 9]], [47, [9, 8]],
%e A255247   [71, [21, 16]], [73, [13, 11]], [79, [11, 10]],
%e A255247   [89, [19, 15]], [97, [25, 19]], [103, [17, 14]],
%e A255247   [113, [15, 13]], [127, [29, 22]],
%e A255247   [137, [21, 17]], [151, [19, 16]],
%e A255247   [167, [15, 14]], [191, [31, 24]],
%e A255247   [193, [23, 19]], [199, [37, 28]],
%e A255247   [223, [17, 16]], [233, [35, 27]],
%e A255247   [239, [27, 22]], [241, [41, 31]], ...
%e A255247 n = 1: 5^2 - 2*4^2 = 25 - 32 = -7 = -A001132(1).
%e A255247 a(3) = -(3*3 - 4*4) = 16 - 9 = 7.
%Y A255247 Cf. A001132, A255248, A255235, A255246, A254936, A255233, A254930.
%K A255247 nonn
%O A255247 1,1
%A A255247 _Wolfdieter Lang_, Feb 19 2015
%E A255247 More terms from _Colin Barker_, Feb 26 2015