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

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  • Phil Carmody
    ... Use calc instead. Or bc. Or the other calc . Or use gp and use p=nextprime(p+1) rather than forprime(p= . ... One of (p-1) and (p+1) is divisible by
    Message 1 of 20 , Jan 3, 2003
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      --- Jon Perry <perry@...> wrote:
      > It was actually a fish for some voluntary labour. Pari chokes on
      > primes>1000000, and I have no other means of performing such high powers
      > tests that would be required, let alone access to a modern machine.

      Use 'calc' instead. Or bc. Or the other 'calc'. Or use gp and use
      'p=nextprime(p+1)' rather than 'forprime(p='.

      > I did check it over a range of values, and my bounds were deduced from these
      > tests. As phi(n)/n has no limit, I would assume this doesn't either, but I
      > was surprised by the narrow region of results.

      One of (p-1) and (p+1) is divisible by 2,
      the other is divisible by 4 or 2^i i>2

      One of (p-1) and (p+1) is divisible by 3.


      The 2s combine such that

      Phi((p-1)*(p+1)) = Phi(2.2^i.(p-1)/2.(p+1)/2^i)
      = 2^i.Phi((p-1)(p+1)/2^(i+1))

      The factor of three gives you a 2/3 factor.


      Therefore the highest value you will find will be
      1/2*2/3 = 1/3 from p=3,5,17
      and 1/3-eps from numbers with a few prime factors larger than 2 or 3 in p+1
      and p-1.

      e.g.
      499637 0.3333266618578673045764089881

      (23:01) gp > factor(499637-1)
      %3 =
      [2 2]
      [124909 1]
      (23:02) gp > factor(499637+1)
      %4 =
      [2 1]
      [3 1]
      [83273 1]

      Note that by HL it will reach 1/3-eps infinitely often

      Lower bound - anyone care for a stab? There should be some bound somewhere,
      I'm sure.

      Phil




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    • David Broadhurst <d.broadhurst@open.ac.u
      Jon Perry mistakenly claimed that ... Invoke it with -p10000000 to precompute primes to 10M. Limit is about 430M, as I recall, but then you need to allocate
      Message 2 of 20 , Jan 3, 2003
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        Jon Perry mistakenly claimed that

        > Pari chokes on primes>1000000

        Invoke it with -p10000000 to precompute primes to 10M.
        Limit is about 430M, as I recall, but then you
        need to allocate core with the -s<size> modifier.
      • David Broadhurst <d.broadhurst@open.ac.u
        ... .......................? No. p=3 gives 1/2 and p=2 gives 2/3 which is maximal, for prime p.
        Message 3 of 20 , Jan 3, 2003
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          Phil:
          > Therefore the highest value you will find will be
          > 1/2*2/3 = 1/3 from p=3,5,17
          .......................?
          No. p=3 gives 1/2
          and p=2 gives 2/3 which is maximal, for prime p.
        • Phil Carmody
          ... Deliberate mistake, to see if Jon was paying attention! ;-) Phil (lying through his teeth!) ===== The answer to life s mystery is simple and direct: Sex
          Message 4 of 20 , Jan 3, 2003
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            --- "David Broadhurst <d.broadhurst@...>" <d.broadhurst@...> wrote:
            > Phil:
            > > Therefore the highest value you will find will be
            > > 1/2*2/3 = 1/3 from p=3,5,17
            > .......................?
            > No. p=3 gives 1/2
            > and p=2 gives 2/3 which is maximal, for prime p.

            Deliberate mistake, to see if Jon was paying attention! ;-)

            Phil
            (lying through his teeth!)


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          • David Broadhurst <d.broadhurst@open.ac.u
            ... 0. We believe (but cannot prove) that there are an infinite number of primes of the form primorial+1. That would be enough to make f(p)=phi(p^2-1)/(p^2-1)
            Message 5 of 20 , Jan 3, 2003
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              Phil:
              > Lower bound - anyone care for a stab?

              0.

              We believe (but cannot prove)
              that there are an infinite number of primes of
              the form primorial+1. That would be enough
              to make f(p)=phi(p^2-1)/(p^2-1)
              as close to zero as one likes.

              At present we know that
              p=392113#+1 is prime,
              giving (a la Mertens)
              f(p) < 0.0436

              Can anyone get lower than that?

              David
            • David Broadhurst <d.broadhurst@open.ac.u
              Let f(p)=phi(p^2-1)/(p^2-1). Say a prime p is lowest yet if there is no prime q
              Message 6 of 20 , Jan 3, 2003
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                Let f(p)=phi(p^2-1)/(p^2-1).
                Say a prime p is "lowest yet" if there is
                no prime q<p with f(q)<f(p).
                The "lowest yet" sequence begins
                2, 3, 5, 11, 29, 131, 139, 181, 419, 1429, 17291, 23561,
                23869, 188189, 315589, 483209, 614041, 1624349, 1729001,
                8242961, 15431989, 22486309, 27033161, 36058021, 57762431,
                61577671, 117048931, ...

                (117048931^2-1)/4=
                2*3*5*7*11*13*19*23*29*31*41*73*97

                What comes next?
              • Phil Carmody
                ... 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
                Message 7 of 20 , Jan 3, 2003
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                  --- "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


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                • Phil Carmody
                  ... Nice concrete follow-on from Jon s original. I can t see a clever way to improve on brute-force search without leaving the possibility of missing some.
                  Message 8 of 20 , Jan 3, 2003
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                    --- "David Broadhurst <d.broadhurst@...>" <d.broadhurst@...> wrote:
                    > Let f(p)=phi(p^2-1)/(p^2-1).
                    > Say a prime p is "lowest yet" if there is
                    > no prime q<p with f(q)<f(p).
                    > The "lowest yet" sequence begins
                    > 2, 3, 5, 11, 29, 131, 139, 181, 419, 1429, 17291, 23561,
                    > 23869, 188189, 315589, 483209, 614041, 1624349, 1729001,
                    > 8242961, 15431989, 22486309, 27033161, 36058021, 57762431,
                    > 61577671, 117048931, ...
                    >
                    > (117048931^2-1)/4=
                    > 2*3*5*7*11*13*19*23*29*31*41*73*97
                    >
                    > What comes next?

                    Nice concrete follow-on from Jon's original.

                    I can't see a clever way to improve on brute-force search without leaving
                    the possibility of missing some.
                    However, it might be possible to stick some markers in the ground for people
                    to aim at by finding squarefree-smooths that have isprime(sqrt(4s+1)).


                    Phil


                    =====
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                  • richard_heylen <richard_heylen@yahoo.co.
                    ... ... I believe it continues as follows 181333151 267190769 331413809 376754951 636510601 1737265531 3019962791 One can obtain fairly low values of
                    Message 9 of 20 , Jan 3, 2003
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                      --- In primenumbers@yahoogroups.com, "David Broadhurst
                      <d.broadhurst@o...>" <d.broadhurst@o...> wrote:
                      > Let f(p)=phi(p^2-1)/(p^2-1).
                      > Say a prime p is "lowest yet" if there is
                      > no prime q<p with f(q)<f(p).
                      > The "lowest yet" sequence begins

                      <snip>

                      > 61577671, 117048931, ...

                      I believe it continues as follows
                      181333151
                      267190769
                      331413809
                      376754951
                      636510601
                      1737265531
                      3019962791

                      One can obtain fairly low values of f(p) realtively easily. Consider
                      for example
                      p=58531393985146662592474024598667898081212671 prime
                      p-1=2.5.7.13.19.43.53.67.71.73.43520821168673.98287085283258329
                      p+1=2^8.3^3.11.17.23.29.31.37.41.47.59.61.79.83.89.97.101.103.107.109.
                      113.127.131.661

                      So the first 33 primes are factors of p^2-1 and I believe this gives
                      an f(p) around 0.113
                      This is significantly smaller than the f(p) around 0.148 for the best
                      of the minimal examples listed.
                      To break the 0.10 barrier you need primes up to 257.
                      To break the 0.05 barrier you need primes up to 75029
                      By this stage, the numbers are getting rather large.

                      Richard Heylen
                    • David Broadhurst <d.broadhurst@open.ac.u
                      ... I proved the first 4 of your addenda with the unsmart brute-force Pari-GP source m=1;mp=430*10^6; Jon please note
                      Message 10 of 20 , Jan 4, 2003
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                        Richard:

                        > I believe it continues as follows
                        > 181333151
                        > 267190769
                        > 331413809
                        > 376754951
                        > 636510601
                        > 1737265531
                        > 3019962791

                        I proved the first 4 of your addenda with
                        the unsmart brute-force Pari-GP source

                        m=1;mp=430*10^6; \\ Jon please note
                        forprime(p=2,mp,n=p^2-1;s=eulerphi(n)/n;if(s<m,m=s;print(p)))

                        David
                      • Jon Perry
                        m=1;mp=430*10^6; Jon please note forprime(p=2,mp,n=p^2-1;s=eulerphi(n)/n;if(s
                        Message 11 of 20 , Jan 4, 2003
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                          'm=1;mp=430*10^6; \\ Jon please note
                          forprime(p=2,mp,n=p^2-1;s=eulerphi(n)/n;if(s<m,m=s;print(p)))'

                          I'm looking...

                          'Use 'calc' instead. Or bc. Or the other 'calc'. Or use gp and use
                          'p=nextprime(p+1)' rather than 'forprime(p='.'

                          Is this the K.R. Matthews Number Theory calculator 'calc'?

                          Is there such a concept as the 'average value of f(p)'?

                          Jon Perry
                          perry@...
                          http://www.users.globalnet.co.uk/~perry/maths/
                          http://www.users.globalnet.co.uk/~perry/DIVMenu/
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                        • David Broadhurst <d.broadhurst@open.ac.u
                          ... forprime is faster than nextprime if you can afford the memory up to p=430M
                          Message 12 of 20 , Jan 4, 2003
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                            > use 'p=nextprime(p+1)' rather than 'forprime(p='
                            'forprime' is faster than 'nextprime'
                            if you can afford the memory up to p=430M
                          • Phil Carmody
                            ... Possibly. I m using Chongo s calc (Curt Landon Noll, record prime finder 2-3 decades ago), which is the standard GNU utility. The whereabouts of the
                            Message 13 of 20 , Jan 4, 2003
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                              --- Jon Perry <perry@...> wrote:
                              > 'm=1;mp=430*10^6; \\ Jon please note
                              > forprime(p=2,mp,n=p^2-1;s=eulerphi(n)/n;if(s<m,m=s;print(p)))'
                              >
                              > I'm looking...
                              >
                              > 'Use 'calc' instead. Or bc. Or the other 'calc'. Or use gp and use
                              > 'p=nextprime(p+1)' rather than 'forprime(p='.'
                              >
                              > Is this the K.R. Matthews Number Theory calculator 'calc'?

                              Possibly. I'm using Chongo's calc (Curt Landon Noll, record prime finder 2-3
                              decades ago), which is the standard 'GNU' utility. The whereabouts of the
                              other calc is answered in the archives some time around a year back, maybe
                              more.

                              > Is there such a concept as the 'average value of f(p)'?

                              I expect it to drift downards so it's not well-defined.
                              (or maybe it is, maybe it's zero. On average numbers have 1/eps distinct
                              divisors, i.e. a divergent number. That's got to take a toll on the phi
                              value. Any sample up to 300000# is puny compared with the sizes of almost
                              all integers...)

                              Phil


                              =====
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                              Sex and death. -- Ian 'Lemmy' Kilminster

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                            • Jon Perry
                              I expect it to drift downards so it s not well-defined. (or maybe it is, maybe it s zero. On average numbers have 1/eps distinct divisors, i.e. a divergent
                              Message 14 of 20 , Jan 4, 2003
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                                'I expect it to drift downards so it's not well-defined.
                                (or maybe it is, maybe it's zero. On average numbers have 1/eps distinct
                                divisors, i.e. a divergent number. That's got to take a toll on the phi
                                value. Any sample up to 300000# is puny compared with the sizes of almost
                                all integers...)'

                                'Therefore the highest value you will find will be
                                1/2*2/3 = 1/3 from p=3,5,17
                                and 1/3-eps from numbers with a few prime factors larger than 2 or 3 in p+1
                                and p-1.'

                                Cough, cough. You make these up, or do they come naturally?

                                Jon Perry
                                perry@...
                                http://www.users.globalnet.co.uk/~perry/maths/
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                              • Phil Carmody
                                ... Where the 3 was already pointed out as a typo. ... OK John. Which of the two statements do you think is wrong? And why? Come on, show us the flaws, I yearn
                                Message 15 of 20 , Jan 4, 2003
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                                  --- Jon Perry <perry@...> wrote:

                                  Quoting me:

                                  > 'I expect it to drift downards so it's not well-defined.
                                  > (or maybe it is, maybe it's zero. On average numbers have 1/eps distinct
                                  > divisors, i.e. a divergent number. That's got to take a toll on the phi
                                  > value. Any sample up to 300000# is puny compared with the sizes of almost
                                  > all integers...)'

                                  > 'Therefore the highest value you will find will be
                                  > 1/2*2/3 = 1/3 from p=3,5,17
                                  > and 1/3-eps from numbers with a few prime factors larger than 2 or 3 in p+1
                                  > and p-1.'

                                  Where the 3 was already pointed out as a typo.

                                  > Cough, cough. You make these up, or do they come naturally?

                                  OK John. Which of the two statements do you think is wrong?
                                  And why?

                                  Come on, show us the flaws, I yearn to be enlightened by your razer-sharp
                                  mathematical quill. I'll even fill in the ellipses, if you like, as have a
                                  feeling you're getting confused by my elision.

                                  �10 to Oxfam for each statement you persuade me to retract. I trust you'll
                                  reciprocate?

                                  Phil


                                  =====
                                  The answer to life's mystery is simple and direct:
                                  Sex and death. -- Ian 'Lemmy' Kilminster

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                                • Jon Perry
                                  £10 to Oxfam for each statement you persuade me to retract. I trust you ll reciprocate? What... Oxfam will pay me £10 to persuade you to retract them? I
                                  Message 16 of 20 , Jan 4, 2003
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                                    '£10 to Oxfam for each statement you persuade me to retract. I trust you'll
                                    reciprocate?'

                                    What... Oxfam will pay me £10 to persuade you to retract them?

                                    I believe they are both correct, hence I will not allow Oxfam to waste their
                                    money on me, and this in turn leads me to believe that f(p) could have an
                                    average value.

                                    Jon Perry
                                    perry@...
                                    http://www.users.globalnet.co.uk/~perry/maths/
                                    http://www.users.globalnet.co.uk/~perry/DIVMenu/
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