Re: Game: find the integers

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

On Wed, Nov 06, 2024 at 02:41:42PM -0800, American Citizen wrote:
> This strongly suggests running the following GP-Pari script
> 
> for(i=0,100,for(j=1,10000000,n=j*(2^(2*i))-1;k=2*eulerphi(n)-n-1;if(k==0,print(2*i,",",j,",",n))))
> 
> but I am not sure of how much time this is going to take.

At least it finds the first non trivial solution (83623935) quickly:

? for(i=0,100,for(j=1,10000000,n=j*(2^(2*i))-1;k=2*eulerphi(n)-n-1;if(k==0,print(2*i,",",j,",",n))))
0,2,1
0,4,3
0,16,15
0,256,255
0,65536,65535
2,1,3
2,4,15
2,64,255
2,16384,65535
4,1,15
4,16,255
4,4096,65535
4,5226496,83623935

>From that it is easy to get the missing ones, so good answer!

Cheers,
Bill