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Re: sum of the squares of primes equalling a square

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  • Mark
    True. And those identities from William were pretty neat. I had said, If the sum of the squares of 5 primes equals a square, two of the primes are 2 and 3
    Message 1 of 7 , Jul 6, 2012
      True. And those identities from William were pretty neat.

      I had said,

      "If the sum of the squares of 5 primes equals a square, two of the primes are 2
      and 3"

      which was a little vague. It should really have been

      If the sum of the squares of 5 primes equals a square, one of the primes is 2 and another is 3.

      Besides the previous

      If the sum of the squares of 7 primes equals a square, three of the primes are
      2.

      we now have this:

      If the sum of the squares of 11 primes equals a square, two of the primes are 2 and another is 3.

      I never would have guessed off the top of my head that with this many squares there would be such restrictions!

      Mark



      --- In primenumbers@yahoogroups.com, Jack Brennen <jfb@...> wrote:
      >
      > Yes, the sum of squares of 3 composites can equal a square.
      >
      > But Mark stated that the sum of squares of 3 primes cannot;
      > it's pretty easy to prove. (Choose the right modulus for
      > solving your Diophantine equation and it becomes clear.)
      >
      > On 7/6/2012 8:53 AM, elevensmooth wrote:
      > >
      > >
      > > --- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@> wrote:
      > >>
      > >> Some tidbits that some may find interesting and perhaps fun to prove:
      > >>
      > >> The sum of the squares of 3 primes never equals a square.
      > >
      > > 8^2 + 49^2 + 64^2 = 81^2
      > >
      > > For any integers p, q, s, and t let
      > >
      > > d=(p^2+q^2+s^2+t^2)/2
      > > c=(p^2+q^2-s^2-t^2)/2
      > > a=ps+qt
      > > b=pt-qs
      > >
      > > then a^2 + b^2 + c^2 = d^2
      > >
      > > Depending on the choices, c and d might be halves, and all the values will need be doubled to get integers.
      > >
      > > William Lipp
      > > Poohbah of OddPerfect.org
      > >
      > >
      > >
      > >
      > > ------------------------------------
      > >
      > > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
      > > The Prime Pages : http://primes.utm.edu/
      > >
      > > Yahoo! Groups Links
      > >
      > >
      > >
      > >
      > >
      >
    • Phil Carmody
      ... Most likely attack is mod 24. Squares are {0,1,4,9,16,12}. Prime squares are {1} and the trivial (choice-free) {4,9} So we have some 2s, some 3s, and then
      Message 2 of 7 , Jul 6, 2012
        --- On Fri, 7/6/12, Mark <mark.underwood@...> wrote:
        > Some tidbits that some may find interesting and perhaps fun to prove:
        >
        > The sum of the squares of 3 primes never equals a square.
        >
        > If the sum of the squares of 5 primes equals a square, two
        > of the primes are 2 and 3.
        >
        > (For example 2^2 + 3^2 + 5^2 + 7^2 + 13^2 = 16^2)

        Most likely attack is mod 24.
        Squares are {0,1,4,9,16,12}.
        Prime squares are {1} and the trivial (choice-free) {4,9}

        So we have some 2s, some 3s, and then some things that square to 1:

        \ 0*2 1*2 2*2 3*2 4*2 5*2
        0*3 0+5 4+4 8+3 12+2 16+1 20+0 | no solns
        1*3 9+4 13+3 17+2 21+1 1+0 x | 13+3 and 1+0 are squares
        2*3 18+3 22+2 2+1 6+0 x x | 22+2 is a square
        3*3 3+2 7+1 11+0 x x x | no solns
        4*3 12+1 16+0 x x x x | 16+0 is a square
        5*3 21+0 x x x x x | no solns

        13+3 is {2, 3, p, q, r}
        1+0 is {2, 2, 2, 2, 3}
        22+2 is {2, 3, 3, p, q}
        16+0 is {2, 3, 3, 3, 3}

        Your result is confirmed.

        > If the sum of the squares of 7 primes equals a square, three
        > of the primes are 2.

        Presuming the above attack works too:


        \ 0*2 1*2 2*2 3*2 4*2 5*2 6*2 7*2
        0*3 0+7 4+6 8+5 12+4 16+3 20+2 0+2 4+2 | 12+4 has 3*2
        1*3 9+6 13+5 17+4 21+3 1+2 5+1 9+0 x | 21+3 and 9+0 have >=3*2
        2*3 18+5 22+4 2+3 6+2 10+1 14+0 x x | no solns
        3*3 3+4 7+3 11+2 15+1 19+0 x x x | 15+1 has 3*2
        4*3 12+3 16+2 20+1 0+0 x x x x | 0+0 has 3*2
        5*3 21+2 1+1 5+0 x x x x x | no solns
        6*3 6+1 10+0 x x x x x x | no solns
        7*3 15+0 x x x x x x x | no solns

        Confirmed.

        Whenever there's a puzzle involving squares, 24 is a very powerful lever.

        Phil
      • djbroadhurst
        ... Phil wisely chose to use 24 = 2^3 * 3 for a modular exhaustion. When I first encountered number theory, it took me a while to understand why working mod 8
        Message 3 of 7 , Jul 6, 2012
          --- In primenumbers@yahoogroups.com,
          Phil Carmody <thefatphil@...> wrote:

          > Most likely attack is mod 24.
          ....
          > Your result is confirmed.

          Phil wisely chose to use 24 = 2^3 * 3
          for a modular exhaustion.

          When I first encountered number theory,
          it took me a while to understand why
          working mod 8 was important, when it seemed
          to me (wrongly) that we need only care
          about mod 2.

          If one will glance, for example, at
          http://en.wikipedia.org/wiki/Kronecker_symbol
          then one will see that mod 8 criteria really matter.
          [Also when trying to extract modular square roots.]

          Silly summary: when dealing with any even prime,
          it may be wise to work modulo its cube.

          David
        • Mark
          ... Thank you Phil, that was very helpful. Now if it could only level a house made from hypothetical perfect euler bricks! (That happens to be the problem
          Message 4 of 7 , Jul 7, 2012
            > Whenever there's a puzzle involving squares, 24 is a very powerful lever.

            Thank you Phil, that was very helpful. Now if it could only level a house made from hypothetical perfect euler bricks! (That happens to be the problem that led me down this particular sum of squares of primes theme, albeit as a side diversion!)


            Mark


            --- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...> wrote:
            >
            > --- On Fri, 7/6/12, Mark <mark.underwood@...> wrote:
            > > Some tidbits that some may find interesting and perhaps fun to prove:
            > >
            > > The sum of the squares of 3 primes never equals a square.
            > >
            > > If the sum of the squares of 5 primes equals a square, two
            > > of the primes are 2 and 3.
            > >
            > > (For example 2^2 + 3^2 + 5^2 + 7^2 + 13^2 = 16^2)
            >
            > Most likely attack is mod 24.
            > Squares are {0,1,4,9,16,12}.
            > Prime squares are {1} and the trivial (choice-free) {4,9}
            >
            > So we have some 2s, some 3s, and then some things that square to 1:
            >
            > \ 0*2 1*2 2*2 3*2 4*2 5*2
            > 0*3 0+5 4+4 8+3 12+2 16+1 20+0 | no solns
            > 1*3 9+4 13+3 17+2 21+1 1+0 x | 13+3 and 1+0 are squares
            > 2*3 18+3 22+2 2+1 6+0 x x | 22+2 is a square
            > 3*3 3+2 7+1 11+0 x x x | no solns
            > 4*3 12+1 16+0 x x x x | 16+0 is a square
            > 5*3 21+0 x x x x x | no solns
            >
            > 13+3 is {2, 3, p, q, r}
            > 1+0 is {2, 2, 2, 2, 3}
            > 22+2 is {2, 3, 3, p, q}
            > 16+0 is {2, 3, 3, 3, 3}
            >
            > Your result is confirmed.
            >
            > > If the sum of the squares of 7 primes equals a square, three
            > > of the primes are 2.
            >
            > Presuming the above attack works too:
            >
            >
            > \ 0*2 1*2 2*2 3*2 4*2 5*2 6*2 7*2
            > 0*3 0+7 4+6 8+5 12+4 16+3 20+2 0+2 4+2 | 12+4 has 3*2
            > 1*3 9+6 13+5 17+4 21+3 1+2 5+1 9+0 x | 21+3 and 9+0 have >=3*2
            > 2*3 18+5 22+4 2+3 6+2 10+1 14+0 x x | no solns
            > 3*3 3+4 7+3 11+2 15+1 19+0 x x x | 15+1 has 3*2
            > 4*3 12+3 16+2 20+1 0+0 x x x x | 0+0 has 3*2
            > 5*3 21+2 1+1 5+0 x x x x x | no solns
            > 6*3 6+1 10+0 x x x x x x | no solns
            > 7*3 15+0 x x x x x x x | no solns
            >
            > Confirmed.
            >
            > Whenever there's a puzzle involving squares, 24 is a very powerful lever.
            >
            > Phil
            >
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