Re: Game: find the integers

[American Citizen <website.reader3@gmail.com>]

 n =  ((1 - q)*p + q)/(1-p-q) is one possible answer here for two 
primes p,q.

On 11/8/24 12:33, Bill Allombert wrote:
> On Fri, Nov 08, 2024 at 09:07:11PM +0100, Michael Hortmann wrote:
> > So you probably call 4294967295 a trivial solution.
> Well, if you let this script run a bit longer, you will get it too.
>
> But the trick is as follow:
> Let n be a solution and p a prime number, then n*p is a solution if and only if
>
> 2*phi(n)*(p-1) = n*p+1
>
> which leads to
> (n+1)*(p-1) = n*p+1
> and
> p = n+2
>
> so if n is a solution and n+2 is prime, n*(n+2) is also a solution.
> Starting with 1, we get
>
> 3, 15, 255, 65535, 4294967295
>
> but we can improve this by taking two primes number p and q,
> then n*p*q is a solution if and only if
> 2*phi(n)*(p-1) = n*p*q+1
> which leads to
> (n+1)*(p-1)*(q-1) = n*p*q+1
> which can also be solved in integers (how ?).
>
> This allows to find the other solutions by starting from 1.
>
> Cheers,
> Bill