

A339881


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


2



0, 1, 3, 13, 59, 23, 221, 9, 31, 103, 8807, 8005, 2047, 527593, 15, 1917, 11759, 9409, 52778687, 801, 113759383, 16437, 21, 1275, 305987, 67, 286025, 12656129, 261, 13458244873, 1381, 719175577, 1410305, 77, 13041, 5580152383, 313074529583, 186079, 1175615653, 949, 1434867510253, 186757799729, 11127596791, 116231
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OFFSET

1,3


COMMENTS

The corresponding y values are given in A339882.
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.
There are no improper solutions (with g >= 2, g^2 does not divide 2).
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).
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.
This family is selfconjugate (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].
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)).
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).
See the W. Lang link Table 1 for some principal reduced forms F_{p(n)} (there for p(n) in the Dcolumn, and F_p is called FR(n)) with their ttuples, 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).
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)}.


REFERENCES

D. A. Buell, Binary Quadratic Forms, Springer, 1989.


LINKS

Table of n, a(n) for n=1..44.
Wolfdieter Lang, Cycles of reduced Pell forms, general Pell equations and Pell graphs


FORMULA

Generalized Pell equation: Positive fundamental a(n), with a(n)^2  A045339(n)*A339882(n)^2 = 2, for n >= 1.


EXAMPLE

The fundamental solutions [A045339(n), [x = a(n), y = A339882(n)]] begin:
[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]], ...


CROSSREFS

Cf. A000194, A000037, A000194, A007520, A045339, A307372, A324252, A339882 (y values), A336793 (record y values), A336792 (corresponding odd p numbers).
Sequence in context: A151320 A151227 A151228 * A336791 A268596 A199297
Adjacent sequences: A339878 A339879 A339880 * A339882 A339883 A339884


KEYWORD

nonn


AUTHOR

Wolfdieter Lang, Dec 22 2020


STATUS

approved



