Re: Finding Closed Form Repesentations from Truncated Decimal Expansions

[Karim Belabas <Karim.Belabas@math.u-bordeaux.fr>]

* kevin lucas [2018-01-27 22:44]:
> Hello all,
> 
> I sometimes come across papers or talks in which PARI/GP is said to have
> been used to establish or conjecture complicated integer relations, often
> with a handwavy reference to LLL. I cannot, however, find an explicit
> demonstration for even simple algebraic closed forms like (1+sqrt(5))/2.
> How, for instance, could PARI find for
> 0.22004376711264303785068975981048665667...  the closed form
> (1+sqrt(2))^2/(2^(9/4)*Pi^(3/2))?
> Secondly, how can one incorporate more
> exotic constants into the process, like multiple zeta values or values of
> certain L-functions? Any help or references would of course be highly
> appreciated.

You probably want to look at lindep() and algdep(); algdep recognizes algebraic
numbers, lindep finds "small" linear relations between arbitrary constants.

It helps a lot to renormalize first so as to recognized the simplest possible
numbers:

(22:51) gp > 0.22004376711264303785068975981048665667 * Pi^(3/2) * 2^(9/4)
%1 = 5.8284271247461900976033774484193961569
(22:51) gp > algdep(%,2)
%2 = x^2 - 6*x + 1

(22:51) gp > 0.22004376711264303785068975981048665667 * Pi^(3/2) 
%3 = 1.2252758689416464406320940924551491289
(22:54) gp > algdep(%,8) \\ still works but a degree 8 number is harder...
%5 = 512*x^5 - 768*x^3 - x

(22:52) gp > vector(8, i, suminf(k=0, 16^-k / (8*k+i)))
%4 = [1.0071844764146762286447601474504384966, 0.50647687666743048095593942496123862718, 0.33923024524519907251558532577879605207, 0.25541281188299534160275704815183096744, 0.20500255763642353394415033621849226687, 0.17131707066649745896673277400009690056, 0.14720193467263502719559812885341132709, 0.12907704227514234334584783136798585625]
(22:52) gp > lindep(concat(%, Pi))
%5 = [-4, 0, 0, 2, 1, 1, 0, 0, 1]~

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]
`