Hi to all,

I conclude the discovery of irreducible a priori `factorised'

closed forms with a bit more theory and an interesting question.

We already have that q ( = 2*p + 1) / (2p)! + 1.

Now if we look for primes of the form q = 3*p + 1 then we can use

Wilson's theorem to obtain (q-1)! == -1 mod q or

(3p)! + 1 == 0 mod q or q / (3p)! + 1

Similarly we can look for primes of the form q = X*p +1 and

the new `factored' form is q / (X*p)! + 1 . We can even look

for primes of the form

q = X*p + Y X, and Y integers. Then the factored form is

q / (X*p + (Y-1))! + 1

So we now have a `large' number of new primes to look for, of the

form q = X*p + Y

and each new record prime of this form (Generalised Sophie Gemain)

will enable a still larger number to be factored `a priori.'

My question is this , can anyone prove there are in fact an

infinite number of primes of the form q = 2*p +1 ? (Sophie

Germain) or indeed of the form q = X*p + Y?

From David Burton's book if p and q are twin primes there is a

formula,

p*q / 4((p-1)! + 1) + p which is a `doubly factored' form

p*q / f(p)

And p and q are twin primes. So Phil Carmody and David Underbakke

can claim a new world record factorisation by plugging their record

twin into this form!

Finally, to end on a lighter note and also with another question and

possibly a new standard. H.E. Rose writes in his famous book

p12 `For a typical 500 digit integer it would take more than a

lifetime to find its factors.' My question is, how long would it

take to factor ANY random 500 digit integer today? Perhaps this can

be called the `Rose500' benchmark in honour of H.E. Rose.

Regards,

Paul Mills,

Kenilworth,

England.