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 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 created short wolframscript from it:
> 
> pi@raspberrypi5:~ $ cat prime.wl
> Solve[k+2==11&&
> w*z+h+j-q==0&&
> (g*k+2*g+k+1)*(h+j)+h-z==0&&
> 16*(k+1)^3*(k+2)*(n+1)^2+1-f^2==0&&
> 2*n+p+q+z-e==0&&
> e^3*(e+2)*(a+1)^2+1-o^2==0&&
> (a^2-1)*y^2+1-x^2==0&&
> 16*r^2*y^4*(a^2-1)+1-u^2==0&&
> n+l+v-y==0&&
> (a^2-1)*l^2+1-m^2==0&&
> a*i+k+1-l-i==0&&
> ((a+u^2*(u^2-a))^2-1)*(n+4*d*y)^2+1-(x+c*u)^2==0&&
> p+l*(a-n-1)+b*(2*a*n+2*a-n^2-2*n-2)-m==0&&
> q+y*(a-p-1)+s*(2*a*p+2*a-p^2-2*p-2)-x==0&&
> z+p*l*(a-p)+t*(2*a*p-p^2-1)-p*m==0,
> {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z},
> Integers]
> pi@raspberrypi5:~ $
> 
> 
> And did run on my overclocked with 3GHz Raspberry Pi5 last night.
> I now stopped it after 10.5h of computation.
> 
> The first line states that I am interested to find integer variable values
> for variables a-z for prime 11. Can this system of Diophantine equations be
> computed with PARI/GP (I have much more powerful AMD 7950X CPU for running
> GP) in reasonable amount of time (perhaps some CPU days)?

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. 
(Beware they change k to k+1, so for you k=9, but for them k=10).
For example the first step is to solve k! =k*g+g+k so g = 329890

Since g is an integer, it follows fdorm the converse to Wilson theorem that k+1 is prime.

The reste of the equation establishes that indeed the formula k! = k*g+g+k hold without using the factorial
symbol, so basically it use complex Lucas sequences to compute the factorial (lemma 2.11) etc.

Cheers,
Bill