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A255233 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).

%I A255233 #18 May 23 2025 01:16:21
%S A255233 5,7,13,9,21,11,17,29,19,15,31,37,17,27,33,23,29,21,41,47,37,23,43,33,
%T A255233 49,55,51,31,41,69,53,29,43,59,35,31,45,61,41,67,85,57,47,63,43,53,35,
%U A255233 75,93,37,71,61,83,47,89,39,73,53,63,79,49,85,69,97,103,109,55,65,47,77,67,83,49
%N A255233 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007522(n), n >= 1 (primes congruent to 7 mod 8).
%C A255233 The corresponding term y = y2(n) of this fundamental solution of the second class of the (generalized) Pell equation x^2 - 2*y^2 = -A007522(n) = -(1 + A139487(n)*8) is given in 2*A255234(n).
%C A255233 For comments and the Nagell reference see A254938.
%H A255233 M. F. Hasler, <a href="/A255233/b255233.txt">Table of n, a(n) for n = 1..1000</a>, May 22 2025
%F A255233 a(n)^2 - 2*(2*A255234(n))^2 = -A007522(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.
%F A255233 a(n) = -(3*A254938(n) - 4*2*A255232(n)), n >= 1.
%e A255233 The first pairs [x2(n), y2(n)] of the fundamental positive solutions of this second class are (the prime A007522(n) appears as first entry):
%e A255233   [7, [5, 4]], [23, [7, 6]], [31, [13, 10]],
%e A255233   [47, [9, 8]], [71, [21, 16]], [79, [11, 10]], [103, [17, 14]], [127, [29, 22]],
%e A255233   [151, [19, 16]], [167, [15, 14]],
%e A255233   [191, [31, 24]], [199, [37, 28]],
%e A255233   [223, [17, 16]], [239, [27, 22]],
%e A255233   [263, [33, 26]], [271, [23, 20]],
%e A255233   [311, [29, 24]], [359, [21, 20]],
%e A255233   [367, [41, 32]], [383, [47, 36]],
%e A255233   [431, [37, 30]], [439, [23, 22]],
%e A255233   [463, [43, 34]], [479, [33, 28]], ...
%e A255233 n= 4: 9^2 - 2*(2*4)^2 = -47 = -A007522(4).
%e A255233 a(4) = -(3*5 - 4*(2*3)) = 24 - 15 = 9.
%o A255233 (PARI) apply( {A255233(n, p=A007522(n))=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1]*[-3,4]~}, [1..88]) \\ The 2nd optional arg allows to directly specify the prime. - _M. F. Hasler_, May 22 2025
%Y A255233 Cf. A007522 (primes == 7 mod 8), A139487 (8k+7 is prime).
%Y A255233 Cf. 2*A255234 (corresponding y2 values), A254938 (x1 values), 2*A255232 (y2 values), A255247, A254936.
%K A255233 nonn,look,easy
%O A255233 1,1
%A A255233 _Wolfdieter Lang_, Feb 18 2015
%E A255233 More terms from _Colin Barker_, Feb 23 2015
%E A255233 Double-checked and extended by _M. F. Hasler_, May 22 2025