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

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  • Phil Carmody
    Jan 3, 2003
      --- "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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