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Re: Re:need help

  • To: mathgroup at smc.vnet.net
  • Subject: [mg87413] Re: [mg87352] Re:[mg87308] need help
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 10 Apr 2008 02:11:04 -0400 (EDT)
  • References: <200804080938.FAA12123@smc.vnet.net>

On 8 Apr 2008, at 18:38, Patrick Klitzke wrote:
> But the problem is, that for some numbers you need more than 4  
> squares:
> 96=9^2+3^2+2^2+1^2+1^2

A very famous theorem of Lagrange says that every positive integer is  
a sum of at most 4 squares. This result can be found in virtually  
every book on number theory (e.g. see K. Chandrasekharan,  
"Introduction to Analytic number theory"). In the case of 96 we have

PowersRepresentations[96, 4, 2]
{{0, 4, 4, 8}}

in other words

96 = 0^2+4^2+4^2+8^2

so in fact you only need 3 squares. Of course if you allow more  
squares you will get more representations:

PowersRepresentations[96, 5, 2]
{{0, 0, 4, 4, 8}, {1, 1, 2, 3, 9}, {1, 1, 3, 6, 7},
    {1, 3, 5, 5, 6}, {2, 2, 4, 6, 6}, {2, 3, 3, 5, 7}}

Also, the question of finding just one representation without  
restriction on length is trivial since you can represent any number as  
a sum of squares of 1's !

Andrzej Kozlowski






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