Re: bnfisintnorm

Charles Greathouse
Case Western Reserve University

On Tue, Feb 7, 2017 at 3:40 AM, Karim Belabas <Karim.Belabas@math.u-bordeaux.fr> wrote:

* Charles Greathouse [2017-02-07 08:55]:
> Given a prime p = 1 mod 3, how can I use bnfisintnorm to find a (the?)
> value of x such that p = x^2 + 3y^2? I don't understand the output format.

0) Just use

qfbsolve(Qfb(1,0,3), p) \\ here, just Cornacchia

1) bnfisintnorm is (more general but) more complicated in this case
because it will solve x^2 - x*y* + y^2 = n and you have to deal with
changes of variables yourself.

2) Another possibility [ a little worse than qfbsolve in this case ]
is thue(x^2+3, p) [ returns all solutions, ]

(09:37) gp > thue(x^2+3,7)
%2 = [[-2, -1], [-2, 1], [2, -1], [2, 1]]

(09:37) gp > p = nextprime(10^1000);
(09:37) gp > thue(x^2+3, p);
time = 957 ms.
(09:37) gp > qfbsolve(Qfb(1,0,3), p);
time = 901 ms.

N.B. Update from master before trying thue() on this one: an oversight
caused it to return half-integers in this case [ until 5 minutes ago].

Cheers,

K.B.
--
Karim Belabas, IMB (UMR 5251) Tel: (+33) (0)5 40 00 26 17
Universite de Bordeaux Fax: (+33) (0)5 40 00 21 23
351, cours de la Liberation http://www.math.u-bordeaux.fr/~kbelabas/
F-33405 Talence (France) http://pari.math.u-bordeaux.fr/ [PARI/GP]
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