Re: Any chance to compute system of Diophantine exquations in 26 variables in GP?

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

On Fri, Jan 26, 2024 at 11:26:43AM +0100, Bill Allombert wrote:
> On Fri, Jan 26, 2024 at 09:53:47AM +0100, hermann@stamm-wilbrandt.de wrote:
> > I used this system of Diophantine equations:
> > https://en.wikipedia.org/wiki/Formula_for_primes#Formula_based_on_a_system_of_Diophantine_equations
> I assume this is this one:
> <https://maa.org/sites/default/files/pdf/upload_library/22/Ford/JonesSatoWadaWiens.pdf>
> 
> If you follow carefully the proof, you should be able to write a program to solve
> it for any small k ! See middle of page 455. 

The issue is that the smallest solution is doubly exponential in k^4,
so you will probably not be able to compute it.

(for us k=10)

we solve 
(3) (2*k)^3*(2*k+2) * (n+1)^2 + 1 = f^2

? my(Q=quadunit((2*k)^3*(2*k+2)*4)); n=imag(Q)-1; f=real(Q);
? n
%41 = 343772642385433639988435123780
? f
%42 = 144220715637070429940775452568001

(at least!)

but then
p=(n+1)^k
q=(p+1)^n

is going to be too large.

Cheers,
Bill.