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[PrimeNumbers] Re: Tuple Twist to Goldbach

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  • mikeoakes2@aol.com
    In a message dated 13/09/03 14:30:26 GMT Daylight Time, ... An integer can be written as the sum of 3 squares iff it is not of the form 4^a*(8*m+7). [Hardy &
    Message 1 of 7 , Sep 13, 2003
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      In a message dated 13/09/03 14:30:26 GMT Daylight Time,
      mark.underwood@... writes:

      > Almost exactly one
      > sixth of numbers cannot be written as the sum of three squares, no
      > matter how large or small! (Of course I have no idea why this
      > relation happens to walk such a fine line.) I wish *something* about
      > primes was this consistent!
      An integer can be written as the sum of 3 squares iff it is not of the form
      4^a*(8*m+7).
      [Hardy & Wright, 5th edn., p. 311.]

      > (As it happens, it appears that all
      > integers can be written as the sum of 4 squares.
      This is the marvellous "Legendre's Theorem" (adjective applies to both:-),
      which is most easily proved using quaternions.
      [HW, p. 303.]

      > Also, about 2/3rds
      > of integers cannot be written as the sum of 2 squares, and this
      > figure seems to slowly increase and may even level off eventually.)
      A number n can be written as the sum of 2 squares iff n = n1^2*n2, where n1
      is arbitrary but n2 has no prime factors = 3 mod 4.
      {HW, p. 299.]

      Mark: the facts you are describing here have not only been known for a long
      time (at least 200 years!) but are also (alas) straying from the "prime
      numbers" constraints of our list. "Alas" because this is an exceedingly beautiful
      area of number theory, which goes by the general name of "Waring's Problem".
      May I encourage those who haven't come across it before to investigate.
      A few months ago, in fact, I made a small contribution to the subject in a
      NMBRTHRY post:-
      http://listserv.nodak.edu/scripts/wa.exe?A2=ind0304&L=nmbrthry&P=R259

      Mike


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