--- "David Broadhurst <

d.broadhurst@...>" <

d.broadhurst@...> wrote:

> Phil:

> > Lower bound - anyone care for a stab?

>

> 0.

It's what I would have guessed, but my brain has begun to stop working in

the last few hours. (e.g. the p=3 -> 0.5 line was on my screen when I typed

p=3 -> 1/3, so I'm really not with it!)

> We believe (but cannot prove)

> that there are an infinite number of primes of

> the form primorial+1.

As many different prime factors as possible, such that

Product[(p-1)/p] could be over as many terms as possible.

> That would be enough

> to make f(p)=phi(p^2-1)/(p^2-1)

> as close to zero as one likes.

Of course.

> At present we know that

> p=392113#+1 is prime,

> giving (a la Mertens)

> f(p) < 0.0436

>

> Can anyone get lower than that?

Not without using a larger number, probably (it's possible, though, as you

can use fewer factors in p-1, and dope p+1 with them instead - all you

need's a few coincidences). However, finding primes of that size is not for

the faint hearted.

Phil

=====

The answer to life's mystery is simple and direct:

Sex and death. -- Ian 'Lemmy' Kilminster

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