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A339881 Fundamental nonnegative solution x(n) of the Diophantine equation x^2 - A045339(n)*y^2 = -2, for n >= 1.

%I A339881 #8 Dec 23 2020 06:18:44
%S A339881 0,1,3,13,59,23,221,9,31,103,8807,8005,2047,527593,15,1917,11759,9409,
%T A339881 52778687,801,113759383,16437,21,1275,305987,67,286025,12656129,261,
%U A339881 13458244873,1381,719175577,1410305,77,13041,5580152383,313074529583,186079,1175615653,949,1434867510253,186757799729,11127596791,116231
%N A339881 Fundamental nonnegative solution x(n) of the Diophantine equation x^2 - A045339(n)*y^2 = -2, for n >= 1.
%C A339881 The corresponding y values are given in A339882.
%C A339881 The Diophantine equation x^2 - p*y^2 = -2, of discriminant Disc = 4*p > 0 (indefinite binary quadratic form), with prime p can have proper solutions (gcd(x, y) = 1) only for primes p = 2 and p == 3 (mod 8) by parity arguments.
%C A339881 There are no improper solutions (with g >= 2, g^2 does not divide 2).
%C A339881 The prime p = 2 has just one infinite family of proper solutions with nonnegative x values. The fundamental proper solutions for p = 2 is (0, 1).
%C A339881 If a prime p congruent to 3 modulo 8, (p(n) = A007520(n)) has a solution then it can have only one infinite family of (proper) solutions with positive x value.
%C A339881 This family is self-conjugate (also called ambiguous, having with each solution (x, y) also (x, -y) as solution). This follows from the fact that there is only one representative parallel primitive form (rpapf), namely F_{pa(n)} = [-2, 2, -(p(n) - 1)/2].
%C A339881 The reduced principal form of Disc(n) = 4*p(n) is F_{p(n)} = [1, 2*s(n), -(p(n) - s(n)^2)], with s(n) = A000194(n'(n)), if p(n) = A000037(n'(n)). The corresponding (reduced) principal cycle has length L(n) = 2*A307372(n'(n)).
%C A339881 The number of all cycles, the class number, for Disc(n) is h(n'(n)) = A324252(n'(n)), Note that in the Buell reference, Table 2B in Appendix 2, p. 241, all Disc(n) <= 4*1051 = 4204 have class number 2, except for p = 443, 499, 659 (Disc = 1772, 1996, 26).
%C A339881 See the W. Lang link Table 1 for some principal reduced forms F_{p(n)} (there for p(n) in the D-column, and F_p is called FR(n)) with their t-tuples, giving the automporphic  matrix Auto(n) = R(t_1) R(t_1) ... R(t_{L(n)}), where R(t) := Matrix([[0, -1], [1, t]]), and the length of the principal cycle L(n) given above, and in Table 2 for CR(n).
%C A339881 To prove the existence of a solution one would have to show that the rpapf F{pa(n)} is properly equivalent to the principal form F_{pa(n)}.
%D A339881 D. A. Buell, Binary Quadratic Forms, Springer, 1989.
%H A339881 Wolfdieter Lang, <a href="/A324251/a324251_2.pdf">Cycles of reduced Pell forms, general Pell equations and Pell graphs</a>
%F A339881 Generalized Pell equation: Positive fundamental a(n), with a(n)^2 - A045339(n)*A339882(n)^2 = -2, for n >= 1.
%e A339881 The fundamental solutions [A045339(n), [x = a(n), y = A339882(n)]] begin:
%e A339881 [2, [0, 1]], [3, [1, 1]], [11, [3, 1]], [19, [13, 3]], [43, [59, 9]], [59, [23, 3]], [67, [221, 27]], [83, [9, 1]], [107, [31, 3]], [131, [103, 9]], [139, [8807, 747]], [163, [8005, 627]], [179, [2047, 153]], [211, [527593, 36321]], [227, [15, 1]], [251, [1917, 121]], [283, [11759, 699]], [307, [9409, 537]], [331, [52778687, 2900979]], [347, [801, 43]], [379, [113759383, 5843427]], [419, [16437, 803]], [443, [21, 1]], [467, [1275, 59]], [491, [305987, 13809]], [499, [67, 3]], [523, [286025, 12507]], [547, [12656129, 541137]], [563, [261, 11]], [571, [13458244873, 563210019]], [587, [1381, 57]], [619, [719175577, 28906107]], [643, [1410305, 55617]], [659, [77, 3]], [683, [13041, 499]], [691, [5580152383, 212279001]], [739, [313074529583, 11516632737]], [787, [186079, 6633]], [811, [1175615653, 41281449]], [827, [949, 33]], [859, [1434867510253, 48957047673]], [883, [186757799729, 6284900361]], [907, [11127596791, 369485787]], [947, [116231, 3777]], ...
%Y A339881 Cf. A000194, A000037, A000194, A007520, A045339, A307372, A324252, A339882 (y values), A336793 (record y values), A336792 (corresponding odd p numbers).
%K A339881 nonn
%O A339881 1,3
%A A339881 _Wolfdieter Lang_, Dec 22 2020