## 10634Re: [PrimeNumbers] Re: Is phi(p^2-1)/(p^2-1) bounded?

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• Jan 3, 2003
> 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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