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Re: [PrimeNumbers] all primes as 1a^2 + 2b^2 +3c^2

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  • mikeoakes2@aol.com
    ... An interesting line of inquiry, Mark. Just 3 points: 1. For these Waring-type problems, the answer as to whether or not a *prime* is representable in a
    Message 1 of 7 , Apr 8, 2006
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      "Mark Underwood" <mark.underwood@...> wrote:

      >It appears likely that every prime can be written as the sum
      >1a^2 + 2b^2 + 3c^2, for a,b,c >= 0.
      >
      >For instance the prime 61 can be expressed in two such ways:
      >
      >61 = 1*4^2 + 2*3^2 + 3*3^2
      >and
      >61 = 1*7^2 + 2*0^2 + 3*2^2.
      >
      >Also, given that a prime p can be written in n ways as
      >a^2 + 2b^c +3c^2, then it can be also written in n ways as
      >2b^2 +3c^2 + 4d^2, where d is either a/2 or (a+1)/2.
      >
      >It looks like all primes can also be written as a^2 + b^2 + 2c^2, as
      >well as a^2 + b^2 + 3c^2. I doubt that there are other ways.


      An interesting line of inquiry, Mark.

      Just 3 points:

      1.
      For these "Waring-type" problems, the answer as to whether or not a *prime* is representable in a particular form is generally the same as to whether or not an *integer* is; so, usually, it is the latter case that is studied.

      2.
      You are presumably familiar with these results:-
      (a) any integer is representable as
      e1*a^2 + e2*b^2 + e3*c^2
      where e1, e2, e3 are in the set {-1,0,+1}.
      (b) any non-negative integer which is *not* of the form 4^k*(8*m+7) is representable as
      e1*a^2 + e2*b^2 + e3*c^2
      where e1, e2, e3 are in the set {0,+1}.
      [See e.g. Hardy & Wright, p. 311.]

      3.
      For some forms, the "product" rule holds; that is, if n1 and n2 are of a given form, then one can show by algebra that n1*n2 is too.
      This holds for the represention
      a^2 + b^2 + c^2 + d^2,
      for example, as one can prove by a tedious but straightforward multiplication exercise.
      I wonder which (if any) of your proposed forms have this property?

      -Mike Oakes
    • mikeoakes2@aol.com
      mark.underwood@sympatico.ca ( Mark Underwood ) wrote:- ... Legendre proved that all positive odd integers can be represented in that form - see e.g. page 3 of:
      Message 2 of 7 , Apr 8, 2006
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        mark.underwood@... ("Mark Underwood") wrote:-

        >It looks like all primes can also be written as a^2 + b^2 + 2c^2

        Legendre proved that all positive odd integers can be represented in that form - see e.g. page 3 of:
        www.math.wisc.edu/~ono/reprints/025.pdf

        Since all even primes can also be so represented, your conjecture has indeed been proved (several centuries ago).

        -Mike Oakes
      • Mark Underwood
        ... in ... has indeed been proved (several centuries ago). ... Yes, a guru has just presented me with the proof. He has also conjectured that 1a^2 + 2b^2 +
        Message 3 of 7 , Apr 8, 2006
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          --- In primenumbers@yahoogroups.com, mikeoakes2@... wrote:
          >
          > mark.underwood@... ("Mark Underwood") wrote:-
          >
          > >It looks like all primes can also be written as a^2 + b^2 + 2c^2
          >
          > Legendre proved that all positive odd integers can be represented
          in
          that form - see e.g. page 3 of:
          > www.math.wisc.edu/~ono/reprints/025.pdf
          >
          > Since all even primes can also be so represented, your conjecture
          has
          indeed been proved (several centuries ago).
          >

          Yes, a guru has just presented me with the proof. He has also
          conjectured that 1a^2 + 2b^2 + 3c^2 itself can represent all odd
          positive integers.

          While I'm at it I want to recant what I said about 2a^2 + 3b^2 +
          4c^2. There was a problem in my code and I see that it certainly
          doesn't cover all the primes.

          Yet it appears that every odd prime can be represented as
          a^2 + 3b^2 + 4c^2.

          Here's one I might add to the Prime Curios:
          43 may be the only prime which cannot be expressed as
          2a^2 + 3b^2 + 5c^2.

          Mark
        • mikeoakes2@aol.com
          ... If we give up the restriction to *odd* integers, it is interesting to find that some (even) numbers can t be represented. This is fully in accord with the
          Message 4 of 7 , Apr 8, 2006
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            "Mark Underwood" <mark.underwood@...> wrote:

            >>>It looks like all primes can also be written as a^2 + b^2 + 2c^2
            >>
            >> Legendre proved that all positive odd integers can be represented
            >>in that form - see e.g. page 3 of:
            >> www.math.wisc.edu/~ono/reprints/025.pdf
            >>
            >> Since all even primes can also be so represented, your conjecture
            >>has indeed been proved (several centuries ago).
            >>
            >
            >Yes, a guru has just presented me with the proof. He has also
            >conjectured that 1a^2 + 2b^2 + 3c^2 itself can represent all odd
            >positive integers.
            >

            If we give up the restriction to *odd* integers, it is interesting to find that some (even) numbers can't be represented.

            This is fully in accord with the remark on p.6 of the following excellent paper:-
            http://www.pubmedcentral.gov/articlerender.fcgi?artid=1076292
            namely: "Universal forms - A positive form representing all positive integers .. is said to be universal. Methods of the previous section show that no positive ternary quadratic form is universal."

            So there were *bound* to be integers not representable.

            Up to 200, the list is just the following:-
            10=2*5 = 2 mod 8
            26=2*13 = 2 mod 8
            40=2^3*5 = 0 mod 8
            42=2*3*7 = 2 mod 8
            58=2*29 = 2 mod 8
            74=2*37 = 2 mod 8
            90=2*3^2*5 = 0 mod 8
            104=2^3*13 = 0 mod 8
            106=2*53 = 2 mod 8
            122=2*61 = 2 mod 8
            138=2*3*23 = 2 mod 8
            154=2*7*11 = 2 mod 8
            160=2^5*5 = 0 mod 8
            168=2^3*3*7 = 0 mod 8
            170=2*7*17 = 2 mod 8
            186=2*3*31 = 2 mod 8

            Note the remarkable restriction to 0 or 2 mod 8 !
            Other than that, there doesn't appear to be any obvious pattern.

            (NB Even Ramanujan was unable to come up with patterns for the exceptions to the ternary forms he studied, so that strongly indicates there aren't any:-)

            (Note that all this is departing more and more from the theme of our list: prime numbers...)

            -Mike Oakes
          • mikeoakes2@aol.com
            ... That was for the form a^2 + 2*b^2 + 3*c^2. It s interesting to explore also the exceptions to the form a^2 + b^2 + 2*c^2 (*) In this case there *does*
            Message 5 of 7 , Apr 10, 2006
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              I wrote:

              >"Mark Underwood" <mark.underwood@...> wrote:
              >
              >>>>It looks like all primes can also be written as a^2 + b^2 + 2c^2
              >>>
              >>> Legendre proved that all positive odd integers can be represented
              >>>in that form - see e.g. page 3 of:
              >>> www.math.wisc.edu/~ono/reprints/025.pdf
              >>>
              >>
              >>Yes, a guru has [...] also
              >>conjectured that 1a^2 + 2b^2 + 3c^2 itself can represent all odd
              >>positive integers.
              >>
              >
              >If we give up the restriction to *odd* integers, it is interesting to find that some (even) numbers can't be represented.
              >
              >This is fully in accord with the remark on p.6 of the following excellent paper:-
              >http://www.pubmedcentral.gov/articlerender.fcgi?artid=1076292
              >namely: "Universal forms - A positive form representing all positive integers .. is said to be universal. Methods of the previous section show that no positive ternary quadratic form is universal."
              >
              >So there were *bound* to be integers not representable.
              >
              >Up to 200, the list is just the following:-
              >10=2*5 = 2 mod 8
              >26=2*13 = 2 mod 8
              >40=2^3*5 = 0 mod 8
              >42=2*3*7 = 2 mod 8
              >58=2*29 = 2 mod 8
              >74=2*37 = 2 mod 8
              >90=2*3^2*5 = 0 mod 8
              >104=2^3*13 = 0 mod 8
              >106=2*53 = 2 mod 8
              >122=2*61 = 2 mod 8
              >138=2*3*23 = 2 mod 8
              >154=2*7*11 = 2 mod 8
              >160=2^5*5 = 0 mod 8
              >168=2^3*3*7 = 0 mod 8
              >170=2*7*17 = 2 mod 8
              >186=2*3*31 = 2 mod 8
              >
              >Note the remarkable restriction to 0 or 2 mod 8 !
              >Other than that, there doesn't appear to be any obvious pattern.


              That was for the form a^2 + 2*b^2 + 3*c^2.

              It's interesting to explore also the exceptions to the form
              a^2 + b^2 + 2*c^2 (*)

              In this case there *does* seem to be a regular pattern, and moreover one which is understandable.

              Consider the form (*) modulo 16.

              Any odd integer has its square = 1 or 9 mod 16,
              any even integer has square = 0 or 4 mod 16.

              So (a^2 + b^2) = {0, 1, 2, 4, 5, 8, 9, 10, 13} mod 16,
              and (2*c^2) = {0, 2, 8} mod 16.

              So (*) <> 14 mod 16.

              Clearly, multiplying a, b and c by 2,
              we have also that
              (*) <> 56 mod 64,
              (*) <> 224 mod 256,
              and so on.

              I have checked up to 760, and there are no other exceptions.

              Would anyone like to check up to some high bound that this is the *complete* set of unrepresetable integers?

              Can anyone *prove* same? [This is likely to be *very* hard!]

              Does anyone know a web page that gives Legendre's proof that all *odd* integers are representable?
              (What a really clever guy he was!)

              -Mike Oakes
            • Jacques Tramu
              ... From: To: Sent: Monday, April 10, 2006 6:52 PM Subject: [PrimeNumbers] Re: all primes as 1a^2 + 2b^2
              Message 6 of 7 , Apr 10, 2006
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                ----- Original Message -----
                From: <mikeoakes2@...>
                To: <primenumbers@yahoogroups.com>
                Sent: Monday, April 10, 2006 6:52 PM
                Subject: [PrimeNumbers] Re: all primes as 1a^2 + 2b^2 +3c^2


                > It's interesting to explore also the exceptions to the form
                > a^2 + b^2 + 2*c^2 (*)
                >
                > (*) <> 56 mod 64,
                > (*) <> 224 mod 256,
                > and so on.
                >
                > I have checked up to 760, and there are no other exceptions.
                >
                > Would anyone like to check up to some high bound that this is the
                > *complete* set of unrepresetable integers?
                >

                I have checked up to 1000000 (10^6) : no other exceptions

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