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Prime pairs.

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  • Prime pairs.

    All positive prime numbers, with the exception of 2 and 3 can be found to lie on two parallel lines. 6x + 5 = y1 and 6x + 7 = y2.

    Now not every number on those two lines are prime. But all the prime numbers, except 2 and 3 in the sequence x=0 , x=1, etc.

    Where x=0, y1=5, y2=7
    x=1, y1=11, y2=13
    x=2, y1=17, y2=19
    etc.
    . . . 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

  • #2
    Originally posted by 37818 View Post
    All positive prime numbers, with the exception of 2 and 3 can be found to lie on two parallel lines. 6x + 5 = y1 and 6x + 7 = y2.
    ... because all numbers which are equivalent to 0/2/4 {mod 6} are divisible by 2, and all numbers which are equivalent to 3 {mod 6} are divisible by three, so all prime numbers except 2 and 3 must be 1 or 5 {mod 6}.
    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


    • #3
      Originally posted by 37818 View Post
      All positive prime numbers, with the exception of 2 and 3 can be found to lie on two parallel lines. 6x + 5 = y1 and 6x + 7 = y2.

      Now not every number on those two lines are prime. But all the prime numbers, except 2 and 3 in the sequence x=0 , x=1, etc.

      Where x=0, y1=5, y2=7
      x=1, y1=11, y2=13
      x=2, y1=17, y2=19
      etc.
      A reasonably well known and provable result. Though normally expressed as 6q +- 1.

      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


      • #4
        Originally posted by Roy View Post
        ... because all numbers which are equivalent to 0/2/4 {mod 6} are divisible by 2, and all numbers which are equivalent to 3 {mod 6} are divisible by three, so all prime numbers except 2 and 3 must be 1 or 5 {mod 6}.
        missed this reply - sorry. About the same summary I was going to give :)

        The statement might look a bit mysterious to those not familiar with primes, but it's not really much different than saying all primes after 2 are odd.

        Jmi
        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


        • #5
          Originally posted by oxmixmudd View Post
          A reasonably well known and provable result. Though normally expressed as 6q +- 1.

          Jim
          Originally posted by oxmixmudd View Post
          missed this reply - sorry. About the same summary I was going to give :)

          The statement might look a bit mysterious to those not familiar with primes, but it's not really much different than saying all primes after 2 are odd.

          Jmi
          Well, 6q - 1 and 6q +1 series are will known. So the two sets of numbers. 5 series incremented by 6 and the 7 series incremented by 6.
          . . . 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


          • #6
            Originally posted by Roy View Post
            ... because all numbers which are equivalent to 0/2/4 {mod 6} are divisible by 2, and all numbers which are equivalent to 3 {mod 6} are divisible by three, so all prime numbers except 2 and 3 must be 1 or 5 {mod 6}.
            Yep.
            . . . 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


            • #7
              Originally posted by 37818 View Post
              Well, 6q - 1 and 6q +1 series are will known. So the two sets of numbers. 5 series incremented by 6 and the 7 series incremented by 6.
              They are the same: -1 ==5 mod 6, 7 ==1 mod 6

              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


              • #8
                Originally posted by oxmixmudd View Post
                They are the same: -1 ==5 mod 6, 7 ==1 mod 6

                Jim
                The mod (6q-1,6) = 5 and mod (6q+1, 6) = 1. So how is that the same?

                Now 6q - 1 series if q = 6ab-a+b those numbers are never prime.
                With 6q + 1 series if q = 6ab-a-b or q = 6ab+a+b those numbers are never prime.
                a and b of course being whole numbers.
                . . . 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


                • #9
                  Originally posted by 37818 View Post
                  The mod (6q-1,6) = 5 and mod (6q+1, 6) = 1. So how is that the same?

                  Now 6q - 1 series if q = 6ab-a+b those numbers are never prime.
                  With 6q + 1 series if q = 6ab-a-b or q = 6ab+a+b those numbers are never prime.
                  a and b of course being whole numbers.
                  OK math basic lesson. What's the big deal?
                  Glendower: I can call spirits from the vasty deep.
                  Hotspur: Why, so can I, or so can any man;
                  But will they come when you do call for them? Shakespeare’s Henry IV, Part 1, Act III:

                  go with the flow the river knows . . .

                  Frank

                  I do not know, therefore everything is in pencil.

                  Comment


                  • #10
                    Originally posted by shunyadragon View Post
                    OK math basic lesson. What's the big deal?
                    Ok. Find a site that explains all primes greater than 3 are found in only in the two number sets. 6q -1 set and 6q + 1 set.
                    Never mind, _here_ is a link to a paper written on this.
                    . . . 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


                    • #11
                      Originally posted by 37818 View Post
                      The mod (6q-1,6) = 5 and mod (6q+1, 6) = 1. So how is that the same?

                      Now 6q - 1 series if q = 6ab-a+b those numbers are never prime.
                      With 6q + 1 series if q = 6ab-a-b or q = 6ab+a+b those numbers are never prime.
                      a and b of course being whole numbers.
                      mod (6q-1,6) = mod (6q+5,6) = 5
                      mod (6q+1,6) = mod (6q+7,6) = 1

                      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


                      • #12
                        Originally posted by 37818 View Post
                        Ok. Find a site that explains all primes greater than 3 are found in only in the two number sets. 6q -1 set and 6q + 1 set.
                        Never mind, _here_ is a link to a paper written on this.
                        Roy already explained it. But perhaps I can reword the explanation a bit so it is perhaps easier to follow.

                        The lines in question define a set of y values (for natural number x values) which might be prime. The reason they do that is that for numbers divided by 6, there are only 6 possible remainders, 0, 1, 2, 3, 4, 5

                        For input values of q that are Natural numbers:

                        The line 6q -1 = y (or for Whole numbers 6q+5) generates all Natural y values with reminder 5 when divided by 6,
                        the line 6q +1 = y (or for Whole numbers 6q+7) generates all Natural y values with remainder 1 when divided by 6.

                        Natural numbers with remainder 0, 2, and 4 are NOT prime because they are all divisible by 2 (because 6 itself is divisible by 2).

                        Natural numbers with remainder 3 are NOT prime because they are divisible by 3 (because 6 is itself divisible by 3)

                        Therefore only numbers divided by 6 that have a remainder of 1 or 5 are possible prime candidates, and thus ALL primes lie on one or the other line.

                        It is not that complicated.


                        But it is interesting. There are many, many more properties that can be associated with primes. It is a fascinating field of number theory.


                        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


                        • #13
                          Merenne numbers 2P - 1 [where P is to be a prime number] all lie only in the 6q + 1 sequence.
                          . . . 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


                          • #14
                            Originally posted by 37818 View Post
                            Now 6q - 1 series if q = 6ab-a+b those numbers are never prime.
                            With 6q + 1 series if q = 6ab-a-b or q = 6ab+a+b those numbers are never prime.
                            a and b of course being whole numbers.
                            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.
                            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


                            • #15
                              Originally posted by 37818 View Post
                              Ok. Find a site that explains all primes greater than 3 are found in only in the two number sets. 6q -1 set and 6q + 1 set.
                              Never mind, _here_ is a link to a paper written on this.
                              That paper appears to be garbage. The author has effectively reinvented the sieve of Erastosthenes and applied it to a misunderstanding of the Riemann zeta function.

                              I stopped skimming it after seeing the author claiming that it was "notable" that the series 6n+1 contains no numbers divisible by 2 or 3.
                              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

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