Another Generalized wilson theorem

[f.firoozbakht at sci.ui.ac.ir]

On Tue, 9 Mar 2004 Mohammed BOUAYOUN <Mohammed.BOUAYOUN at sanef.com> wrote:
 
> Let a(n) = n!.2^n +1
> a(n) is prime for n=1 and n=259 but there are no other prime for
> n<3000 
> Can any one extend this sequence ? 
> I think that there are no other prime, 
> 
> If A(n,b)=((n-1)!.b^(n-1) + 1)/n  and n,b are distinct prime  
>  
> I think that the number of prime A(n,b) is finite
>  
> A(n,2) is prime for n=3,37,61,??
> A(n,3) is prime for n=2,5,??
> A(n,5) is prime for n=2,3,7,13,19,??
> A(n,7) is prime for n=11,17,??
> A(n,11) is prime for n= (no prime n <800) ??
> A(n,13) is prime for n=2,3,??
> A(n,17) is prime for n=3,7,13,??
> A(n,19) is prime for n=3,7,281, ??
>  
>  I remark that if p is prime then p divide a(p-1)
>  
> we can prove this by wilson's theorem and Fermat's Little Theorem
>  
> if p is prime and gcd(q,p)=1 then p divide (p-1)!.q^(p-1)  + 1
> can any one help me if this theorem exist ?
>  
> if q=1 we have wilson's theorem.


Hi Mohammed,

I don't know if your theorem  is known before. 
But this theorem is a nice generalization of wilson's
theorem.

> I think that the number of prime A(n,b) is finite.

The proof or disproof of your conjecture is very difficult.
But I think the number of prime A(n,b) is infinite.
We can extend the definition of A(n,b) to the case where n 
and b are natural numbers,but A(n,b) will be integer if n=1 
or as you wrote if n is prime and gcd (n,b)=1.
So there is no reason to restrict our discussion of A(n,p) to
prime p only.

And we have :

A(1,b) is prime for all b.

A(n,1) is prime for n= 1, 5, 7, 11, 29, 773, ??
A(n,4) is prime for n= 1, 3, 5, 7, ??
A(n,6) is prime for n= 1, 5, 11, 83, ??
A(n,8) is prime for n= 1, 3, 5, 13, ??
A(n,9) is prime for n= 1, 2, 11, ??
A(n,10) is prime for n= 1, 3, 7, ??
A(n,12) is prime for n= 1, 7, 163, ??
A(n,14) is prime for n= 1, 3, ??
A(n,15) is prime for n= 1, 19, ??
A(n,16) is prime for n= 1, (no prime n < 1000) ??
A(n,18) is prime for n= 1, 13, 19, 41, ??
A(n,20) is prime for n= 1, 11 ??
 
 
Best wishes,

Farideh
 
 
 


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