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Factors for the composites of the form: 2^p -1

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  • Peter Lesala
    Is the smallest factor for the composite numbers 2^p - 1 where p is a prime but 2^p - 1 not a prime, always a prime? The theory for the factors of these type
    Message 1 of 12 , Aug 6, 2011
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      Is the smallest factor for the composite numbers 2^p - 1 where p is a prime but 2^p - 1 not a prime, always a prime?

      The theory for the factors of these type of primes has probably been already developed. The theory can be expressed a follows:

      Let p be a prime. If 2^p - 1 is not a Mersenne prime, then it is composite with at least two distinct factors 2*k1*p +1 and 2*k2*p + 1, k1 < k2, and k1, k2 = 1, 2, 3, ...


      An interesting aspect to this theory is whether or not the smallest factor say 2*k1*p is always a prime. The table below illustrates the basics of this theory.


      2^11 - 1 : 89 = 2*4*11 + 1; 23 = 2*11 + 1

      2^23 - 1 : 178 481 = 2*3880*23 + 1; 47 = 2*23 + 1

      2^29 - 1 : 1103 = 2*19*29 + 1; 233 = 2*2*4*29 + 1

      2^37 - 1 : 616 318 177 223 = 2*3*37 + 1

      = 2*8 328 624*37 + 1

      2^41 - 1 :164 511 353 = 4 012 472*41 + 1;

      13 367 = 2*83*41 + 1



      2^43 - 1: 9719 = 2*113*43 + 1; 431 = 2*5*43 + 1

      2^47 - 1 :4513 = 2*48*47 + 1 ;2351 = 2*25*47 + 1

      2^53 -1 6361 = ?; 2*6*53 + 1

      2^59 - 1 : ? ; ?

      2^61 - 1 : ? ; ?

      2^67 - 1 :?; ?

      2^71 - 1 : ? ; ?

      2^73 - 1 :?, 439 = 2*3*73 + 1

      2^79 - 1 : 2687 = 2*2*17*79 + 1

      2^83 - 1 : ?;167 = 2*83 + 1

      2^97 - 1 : ? ;11 447 = 2*59*97 + 1

      2^107 -1 :? ; ?

      2^109 - 1 :? ; ?

      2^113 - 1 : ?; 3391 = 2*15*113 + 1

      2^131 - 1 : ?; 263 = 2*131 + 1

      2^139 - 1: ?;?

      2^149 - 1: ?; ?

      2^151 - 1 : ?; 18 121 = 2*30*151 + 1

      2^157 - 1 :? ; ?

      2^163 - 1 :? ; ?

      2^167 - 1 : ? ; ?

      2^173 - 1 :? ; ?

      2^179 - 1: ?; 359 = 2*179 + 1

      2^181 - 1: ?;4 3441 = 2*120*181 + 1

      2^191 - 1 : ?; 383 = 2*191 + 1

      2^193 - 1 :? ; ?

      2^197 - 1 : ?; 7487 = 2*19*197 + 1

      2^199 - 1 :? ; ?

      2^211 - 1 : ?;15 193 = 2*36*211 + 1

      2^223 - 1 :?/ 18 287 = 2*41*223 + 1


      Peter.

      [Non-text portions of this message have been removed]
    • djbroadhurst
      ... Obviously. For *any* integer N 1, the smallest divisor d|N with d 1 is a prime. Proof: Suppose the contrary. Then there would be a divisor p|d|N with d
      Message 2 of 12 , Aug 6, 2011
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        --- In primenumbers@yahoogroups.com,
        "Peter Lesala" <plesala@...> wrote:

        > Is the smallest factor for the composite numbers 2^p - 1
        > where p is a prime but 2^p - 1 not a prime, always a prime?

        Obviously. For *any* integer N > 1, the smallest divisor
        d|N with d > 1 is a prime.

        Proof: Suppose the contrary. Then there would be a divisor
        p|d|N with d > p > 1. But that is impossible, since d is
        the *smallest* divisor d|N with d > 1.

        David
      • leavemsg1
        Dear Peter Lesala, I remember trying to help you to prove that odd-perfect numbers don t exist. I ve had to abandon many methods, including yours, but was
        Message 3 of 12 , Aug 9, 2011
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          Dear Peter Lesala,

          I remember trying to help you to prove that odd-perfect numbers
          don't exist. I've had to abandon many methods, including yours,
          but was successful. You can view my simple algebraic argument at
          my website... www.oddperfectnumbers.com.

          It boiled down to one equation for all cases and's by induction.
          I think that you'll be pleasantly surprised with my results.

          Best Regards,

          Bill Bouris

          P.S. Sorry for making you upset before... I was just trying to
          improve your paper. My idea definitely works, now.
        • James Merickel
          You re saying you have something almost journal ready proving opn s do not exist? Please announce that publication date. An algebraic proof that odd perfect
          Message 4 of 12 , Aug 9, 2011
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            You're saying you have something almost journal ready proving opn's do not exist? Please announce that publication date. An algebraic proof that odd perfect numbers don't exist must be marvelous reading material. You will get all sorts of lecture invitations too. Great news! Dot com isn't the way to go, though. Looks a little like you are selling bs.
            Jim

            On Tue Aug 9th, 2011 11:54 AM EDT leavemsg1 wrote:

            >Dear Peter Lesala,
            >
            >I remember trying to help you to prove that odd-perfect numbers
            >don't exist. I've had to abandon many methods, including yours,
            >but was successful. You can view my simple algebraic argument at
            >my website... www.oddperfectnumbers.com.
            >
            >It boiled down to one equation for all cases and's by induction.
            >I think that you'll be pleasantly surprised with my results.
            >
            >Best Regards,
            >
            >Bill Bouris
            >
            >P.S. Sorry for making you upset before... I was just trying to
            >improve your paper. My idea definitely works, now.
            >
          • Peter Lesala
            Bill, You really enjoy working at this problem which is good. I have put the past behind me. The group on the Prime Pages is a great team, hence the reason you
            Message 5 of 12 , Aug 14, 2011
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              Bill,

              You really enjoy working at this problem which is good. I have put the past behind me. The group on the Prime Pages is a great team, hence the reason you have had immediate and appropriate feedback all of the time. I wish I could be of more help to you but presently my interest is elsewhere.

              Best of luck.

              Peter.
              ----- Original Message -----
              From: leavemsg1
              To: primenumbers@yahoogroups.com
              Sent: Tuesday, August 09, 2011 5:54 PM
              Subject: [PrimeNumbers] odd-perfect numbers



              Dear Peter Lesala,

              I remember trying to help you to prove that odd-perfect numbers
              don't exist. I've had to abandon many methods, including yours,
              but was successful. You can view my simple algebraic argument at
              my website... www.oddperfectnumbers.com.

              It boiled down to one equation for all cases and's by induction.
              I think that you'll be pleasantly surprised with my results.

              Best Regards,

              Bill Bouris

              P.S. Sorry for making you upset before... I was just trying to
              improve your paper. My idea definitely works, now.





              [Non-text portions of this message have been removed]
            • Roahn Wynar
              Hello all, I am looking for two proofs and don t know where to start. First, I seek the proof that the Gamma function is not the solution of any ordinary
              Message 6 of 12 , Aug 14, 2011
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                Hello all,
                I am looking for two proofs and don't know where to start.

                First, I seek the proof that the Gamma function is not the solution of any ordinary differential equation with rational coeficients. I have seen this proof refered to but have never actually seen it. Is it somehow obvious?

                Also, I am looking for an interesting proof that depends on the Reimann Hypothesis being assumed true. The larger the implications of what ever proof you suggest the better. I would like to use it as an example in a short piece I am writing.

                Thanks in advance!


                Roahn

                [Non-text portions of this message have been removed]
              • Phil Carmody
                ... Riemann. Try to forget the pronunciation of it that you learnt too, as it s almost certainly wrong (and leads to the misspelling). Once you have the
                Message 7 of 12 , Aug 15, 2011
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                  --- On Mon, 8/15/11, Roahn Wynar <rwynar@...> wrote:
                  > I am looking for two proofs and don't know where to start.
                  >
                  > First, I seek the proof that the Gamma function is not the
                  > solution of any ordinary differential equation with rational
                  > coeficients.  I have seen this proof refered to but
                  > have never actually seen it.  Is it somehow obvious?
                  >
                  > Also, I am looking for an interesting proof that depends on
                  > the Reimann

                  Riemann. Try to forget the pronunciation of it that you learnt too, as it's almost certainly wrong (and leads to the misspelling). Once you have the correct pronunciation in your head, the correct spelling should follow automatically.

                  > Hypothesis being assumed true. The larger the
                  > implications of what ever proof you suggest the
                  > better.  I would like to use it as an example in a
                  > short piece I am writing.

                  Maximum gaps between prime numbers http://primes.utm.edu/notes/gaps.html ?
                  Maximum number of tests required for Miller-Rabin to prove primality?

                  I'm not sure there are any "implications", though. As not one single bit will be different in any algorithmic computation no matter if RH is true or false[*]. In the real world, we're usually happy to just say "the probability of X is <= 10^-big, or >= 1-10^-big".

                  You have to remember that the real world is where supposed numerate people (economists) are prepared to pretend the cauchy distribution has a mean and a standard deviation. 91 might as well be prime, and the Riemann Hypothesis might as well be a piece of blank verse, in the face of such ignorance.

                  Phil
                  [* Not strictly true, as RH can tell you to stop earlier, for example, but that basically only changes the bailing out conditions (failures to find something), not the actual numbers you'd rather be finding.]
                • Chris Caldwell
                  ... Phil s example is a fine one. The RH basically says primes are relatively smooth, this allows lots of results that require smoothness. Now it being false
                  Message 8 of 12 , Aug 15, 2011
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                    > Hypothesis being assumed true. The larger the
                    > implications of what ever proof you suggest the
                    > better.  I would like to use it as an example in a
                    > short piece I am writing.

                    Phil's example is a fine one. The RH basically says primes are relatively smooth, this allows lots of results that require smoothness. Now it being false would not falsify many of these results, just the proof it was used as a short cut in. For example, a coauthor and I tried to prove the "obvious" result there is a prime between consecutive cubes so that we could identify the least of the Mills constants. We failed and then replaced 15 pages of mathematics in the paper with a couple sentences based on the RH. It will be proved without RH eventually.

                    If you want things that could be "proved" assuming the RH, but will be false without it, then just Google

                    "equivalent to the Riemann hypothesis"
                  • John
                    Two sent to the Prof meant for the Forum. Pardon me, but that default option for the recipient is crazy, to be blunt! I am not a Prof, but I am absentminded
                    Message 9 of 12 , Aug 16, 2011
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                      Two sent to the Prof meant for the Forum.

                      Pardon me, but that default option for the recipient is crazy, to be blunt!

                      I am not a Prof, but I am absentminded due to a long and abstemious life.

                      --- In primenumbers@yahoogroups.com, Chris Caldwell <caldwell@...> wrote:
                      >
                      > > Hypothesis being assumed true. The larger the
                      > > implications of what ever proof you suggest the
                      > > better.  I would like to use it as an example in a
                      > > short piece I am writing.
                      >
                      > Phil's example is a fine one. The RH basically says primes are relatively smooth, this allows lots of results that require smoothness. Now it being false would not falsify many of these results, just the proof it was used as a short cut in. For example, a coauthor and I tried to prove the "obvious" result there is a prime between consecutive cubes so that we could identify the least of the Mills constants. We failed and then replaced 15 pages of mathematics in the paper with a couple sentences based on the RH. It will be proved without RH eventually.
                      >
                      > If you want things that could be "proved" assuming the RH, but will be false without it, then just Google
                      >
                      > "equivalent to the Riemann hypothesis"
                      >
                    • Chris Caldwell
                      ... Perhaps it is, but I chose it to cut down on accidental emails to the list when one intends to reply just to the recipient. Since such an accidental reply
                      Message 10 of 12 , Aug 16, 2011
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                        > Two sent to the Prof meant for the Forum.
                        > Pardon me, but that default option for the recipient is crazy, to be blunt!

                        Perhaps it is, but I chose it to cut down on accidental emails to the list when one intends to reply just to the recipient. Since such an accidental reply goes to about one thousand folks, it seemed worth the extra cost of having the sender actually look at who they are replying to. Reply all is not much more difficult than reply after all.

                        After you have made this mistake 1000 times, contact me again--that will equal one accidental email to the list. (Maybe it should be after 3,000 times since your e-mail and my reply make up about 2000 emails sent; I replied to the list because other have had this problem--and I want them to all blame me :) )

                        CC
                      • John
                        A good and reasonable answer to my complaint, and I doubt if even I will manage 1000 mistakes. The nub of my query pertains to the obviousness of the case
                        Message 11 of 12 , Aug 16, 2011
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                          A good and reasonable answer to my complaint, and I doubt if even I will manage 1000 mistakes.

                          The nub of my query pertains to the "obviousness" of the case for finding a prime between n^3 and (n+1)^3 (for cubes). I once posited on this forum a similar sentiment concening the harder case of squares, as the heuristics seemed to point to it being "obvious" also.

                          I therefore ask the experts this question if it has a ready answer: Is there a qualitative (rather than quantitative) difference between the case of cubes and that of squares?

                          John


                          --- In primenumbers@yahoogroups.com, Chris Caldwell <caldwell@...> wrote:
                          >
                          > > Two sent to the Prof meant for the Forum.
                          > > Pardon me, but that default option for the recipient is crazy, to be blunt!
                          >
                          > Perhaps it is, but I chose it to cut down on accidental emails to the list when one intends to reply just to the recipient. Since such an accidental reply goes to about one thousand folks, it seemed worth the extra cost of having the sender actually look at who they are replying to. Reply all is not much more difficult than reply after all.
                          >
                          > After you have made this mistake 1000 times, contact me again--that will equal one accidental email to the list. (Maybe it should be after 3,000 times since your e-mail and my reply make up about 2000 emails sent; I replied to the list because other have had this problem--and I want them to all blame me :) )
                          >
                          > CC
                          >
                        • Phil Carmody
                          ... Well there s your problem. This isn t a forum , it is a mailing list that happens to have a web archive of its messages, web message injection capability,
                          Message 12 of 12 , Aug 16, 2011
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                            --- On Tue, 8/16/11, John <mistermac39@...> wrote:
                            > Pardon me, but that default option for the recipient is
                            > crazy, to be blunt!

                            Well there's your problem. This isn't a "forum", it is a mailing list that happens to have a web archive of its messages, web message injection capability, and associated web-based fluff. The reply-to behaviour mimics that of most mail clients since time immemorial. If the web interface makes this choice less convenient one way than the other, then all that says is negative things about the web interface. Which is why it's better to approach it as a mailing list. Chose your mail client, rather than being forced to use yahoo's very clumsy, and sometimes broken, interface.

                            Phil
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