Re: Finding Closed Form Repesentations from Truncated Decimal Expansions

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Sat, Jan 27, 2018 at 10:44:35PM +0100, kevin lucas wrote:
> 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

The way you do it is
? z = 0.22004376711264303785068975981048665667;
? lindep([log(2),log(Pi),log((1+sqrt(2))),log(z)])
%2 = [9,6,-8,4]~
So you get
9*log(2)+6*log(Pi)-8*log(1+sqrt(2))+4*log(z)=0
and then you take the exponential.

Of course you need to use a sufficient accuracy.

Cheers,
Bill.