Hi,

Yves Gallot has pointed out that directly from Wilson's theorem

(p-1)! == -1 mod p so that p divides (p-1)! + 1

(p-1)! +1 is a `closed factorial form' and the generic term I

am probably looking for is an `irreducible' form. (polynomial or

otherwise).

But as Yves pointed out, by substituting the largest known prime

(Mersenne) for p we obtain the p / f(p) World record

p / (p-1)! + 1 . The New World record is that Wilson's theorem

shows that (2^6972593 -2)! + 1 is the largest 'irreducible form'

number for which we have a prime factor.

I approached this problem from Henri Lifchitz's work where he

showed that

q / 3^p 1 . This is a different `factored form' because

it is q / f(p) and q is distinct from p. So Henri's unsung

world record and now my world record still stands too! In fact if

q / f(p) then either q = p or is distinct from p so there are just 2

world records for this factored form! Held by the Mersenne (q = p)

and Sophie Germain (q /= p) WR primes. Now the theory starts, of

course.

Incidentally, it is easy to prove that (2p)! + 1 > 3^p 1 by

induction.

Also I claim to have missed the simple p / f(p) because I was

deflected by the nice formula

(a + b)^p == a^b + b^p mod p

This gives a large number of expressions for the factored form

p / f(p)

i.e (2 + 1)^p == 2^p + 1 mod p => p / 3^p 2^p 1

regards,

Paul Mills

Kenilworth

England.