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Prime pairs.

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  • #16
    Originally posted by 37818 View Post
    Merenne numbers 2P - 1 [where P is to be a prime number] all lie only in the 6q + 1 sequence.
    ...except when P=2, where 2P - 1 = 3.

    Since all primes > 2 are of the form 2n+1,

    2P-1 {mod 6}
    = 22n+1-1 {mod 6}
    = 2(22n)-1 {mod 6}
    = 2(4n)-1 {mod 6}
    = 2((3+1)n)-1 {mod 6}
    = 2(3n + k1.3n-1 + k2.3n-2 + ... + kn-1.3 + 1) -1 {mod 6}, where kx are the coefficients of the binomial expansion of n
    = 2(3.something + 3.something + ... + 3.something + 1) -1 {mod 6}
    = 6.something + 6.something + ... + 6.something + 2 -1 {mod 6}
    = 1 {mod 6}
    Last edited by Roy; 01-11-2018, 07:30 AM.
    Jorge: Functional Complex Information is INFORMATION that is complex and functional.

    MM: First of all, the Bible is a fixed document.
    MM on covid-19: We're talking about an illness with a better than 99.9% rate of survival.

    seer: I believe that so called 'compassion' [for starving Palestinian kids] maybe a cover for anti Semitism, ...

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    • #17
      Originally posted by Roy View Post
      ...except when P=2, where 2P - 1 = 3.

      Since all primes > 2 are of the form 2n+1,

      2P-1 {mod 6}
      = 22n+1-1 {mod 6}
      = 2(22n)-1 {mod 6}
      = 2(4n)-1 {mod 6}
      = 2((3+1)n)-1 {mod 6}
      = 2(3n + k1.3n-1 + k2.3n-2 + ... + kn-1.3 + 1) -1 {mod 6}, where kx are the coefficients of the binomial expansion of n
      = 2(3.something + 3.something + ... + 3.something + 1) -1 {mod 6}
      = 6.something + 6.something + ... + 6.something + 2 -1 {mod 6}
      = 1 {mod 6}
      very nice :)

      Jim
      My brethren, do not hold your faith in our glorious Lord Jesus Christ with an attitude of personal favoritism. James 2:1

      If anyone thinks himself to be religious, and yet does not  bridle his tongue but deceives his own heart, this man’s religion is worthless James 1:26

      This you know, my beloved brethren. But everyone must be quick to hear, slow to speak and slow to anger; James 1:19

      Comment


      • #18
        Originally posted by oxmixmudd View Post
        very nice :)

        Jim
        Both 6q-1 and 6q+1 excluded 2 and 3. So it was to be understood since 22-1 = 3 is outside 6q+1 q beginning with 1.
        . . . the gospel of Christ: for it is the power of God unto salvation to every one that believeth; . . . -- Romans 1:16 KJV

        . . . that Christ died for our sins according to the scriptures; And that he was buried, and that he rose again the third day according to the scriptures: . . . -- 1 Corinthians 15:3-4 KJV

        Whosoever believeth that Jesus is the Christ is born of God: . . . -- 1 John 5:1 KJV

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        • #19
          Originally posted by Roy View Post
          6(6ab-a+b)-1 = 36ab-6a+6b-1 = (6a+1)(6b-1)
          6(6ab-a-b)+1 = 36ab-6a-6b+1 = (6a-1)(6b-1)
          6(6ab+a+b)+1 = 36+6a+6b+1 = (6a+1)(6b+1)

          Therefore they can only be prime if a=0 or b=0.
          If a=0 and b=0 then q=0.
          . . . the gospel of Christ: for it is the power of God unto salvation to every one that believeth; . . . -- Romans 1:16 KJV

          . . . that Christ died for our sins according to the scriptures; And that he was buried, and that he rose again the third day according to the scriptures: . . . -- 1 Corinthians 15:3-4 KJV

          Whosoever believeth that Jesus is the Christ is born of God: . . . -- 1 John 5:1 KJV

          Comment


          • #20
            Originally posted by 37818 View Post
            If a=0 and b=0 then q=0.
            True, but trivial and not really relevant.

            What's your point?
            Jorge: Functional Complex Information is INFORMATION that is complex and functional.

            MM: First of all, the Bible is a fixed document.
            MM on covid-19: We're talking about an illness with a better than 99.9% rate of survival.

            seer: I believe that so called 'compassion' [for starving Palestinian kids] maybe a cover for anti Semitism, ...

            Comment


            • #21
              Originally posted by Roy View Post
              ...except when P=2, where 2P - 1 = 3.

              Since all primes > 2 are of the form 2n+1,

              2P-1 {mod 6}
              = 22n+1-1 {mod 6}
              = 2(22n)-1 {mod 6}
              = 2(4n)-1 {mod 6}

              [and, because 2.4 = 2 mod 6]

              = 1 {mod 6}
              But the binomial expansion was cute.

              Comment


              • #22
                Because the Euler phi function (counting the multiplicative group mod n) is multiplicative, and phi(p) = p-1 for all primes p ...
                phi(6) = phi(2).phi(3) = 1.2 = 2

                Hence, mod 6, all primes > 3 must fall into one of two equivalence classes.


                But no one more than six feet away from a crack pipe would call them paired primes.
                Last edited by Juvenal; 01-15-2018, 08:28 AM.

                Comment


                • #23
                  Originally posted by lao tzu View Post
                  Because the Euler phi function (counting the multiplicative group mod n) is multiplicative, and phi(p) = p-1 for all primes p ...
                  phi(6) = phi(2).phi(3) = 1.2 = 2

                  Hence, mod 6, all primes > 3 must fall into one of two equivalence classes.

                  Nice addition to the conversation



                  Jim
                  My brethren, do not hold your faith in our glorious Lord Jesus Christ with an attitude of personal favoritism. James 2:1

                  If anyone thinks himself to be religious, and yet does not  bridle his tongue but deceives his own heart, this man’s religion is worthless James 1:26

                  This you know, my beloved brethren. But everyone must be quick to hear, slow to speak and slow to anger; James 1:19

                  Comment

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