# Another Generalized wilson theorem

Mohammed BOUAYOUN Mohammed.BOUAYOUN at sanef.com
Tue Mar 9 18:31:59 CET 2004

```Dear SeqFan,

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

Thanks

M.Bouayoun
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