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Proof of Primes of the form 10^(2k+1)+3

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  • Milton Brown
    Here is a proof that 10^5+3 is prime and 10^3+3 is not prime using the fact that 10^(2k+1) + 3 is divisible by a prime P if and only if (P-10)^(2k+1) mod P is
    Message 1 of 4 , Jul 22, 2005
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      Here is a proof that 10^5+3 is prime and
      10^3+3 is not prime using the fact that
      10^(2k+1) + 3 is divisible by a prime P
      if and only if (P-10)^(2k+1) mod P is 3,

      sqrt(10^3+3) = 31.6

      sqrt(10^5+3) = 316.2

      The numbers (P-10)^5 mod P are given
      below for the primes

      2,3,5,7,11,13,17,19,23,29,
      31,37,41,43,47,53,59,61,67,71,
      73,79,83,89,97,101,103,107,109,113,
      127,131,137,139,149,151,157,163,167,173,
      179,181,191,193,197,199,211,223,227,281,
      283,293,307,311,313

      0,1,0,5,19,11,16,4,21,
      6,11,40,18,16,11,5,40,31,39,
      10,14,15,36,7,91,13,45,62,5
      25,121,40,122,23,36,72,86,163,107,
      16,123,30,189,34,57,41,183,97,159,
      219,136,119,211,69,221,21,242,187,273,
      164,59,25,305,160

      Which proves that 10^5+3 is prime since
      no 3 appears in this list.

      For 10^3+3 the numbers are:

      2,3,5,7,11,13,17,19,23,29,31:

      0,1,0,6,1,1,3,7,12,15,23

      which shows that 10^3+3 is divisble by 17.

      Tables of these numbers can be maintained for each
      prime, then divisibility is a matter of table
      look-up. And so, is better than division.
      This can also be done for 10^(2k+1) + 7 or + 9.

      Milton L. Brown
    • Bouk de Water
      Hello everybody! This may seem a stupid question but I m simply not good at making executables ... I need a p4 version of the latest ECM which will run on XP.
      Message 2 of 4 , Aug 2, 2005
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        Hello everybody!

        This may seem a stupid question but I'm simply not good at making executables
        :-)

        I need a p4 version of the latest ECM which will run on XP. I'm currently using
        the old 5.1 for p4 and it keeps crashing all the time on both my machines.

        And could somebody please advise me if I need cygwin or msys to make it run
        properly?

        Help would be very welcome!

        Bouk (who normally let's David almost do all the ECM stuff)



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      • elevensmooth
        ... http://www.mersenneforum.org/showpost.php?p=52303&postcount=105 Stand alone executables.
        Message 3 of 4 , Aug 2, 2005
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          > I need a p4 version of the latest ECM which will run on XP.

          http://www.mersenneforum.org/showpost.php?p=52303&postcount=105

          Stand alone executables.
        • Bouk de Water
          Thanks! Both of you! Bouk. ... __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection
          Message 4 of 4 , Aug 2, 2005
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            Thanks! Both of you!

            Bouk.

            --- elevensmooth <elevensmooth@...> wrote:

            >
            > > I need a p4 version of the latest ECM which will run on XP.
            >
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            >
            > Stand alone executables.
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            > The Prime Pages : http://www.primepages.org/
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