## 982Factor Record

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• Apr 25, 2001
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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.
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