Sums of Three Squares

Date: 05/18/98 at 00:52:59
From: steve darcy
Subject: Sum of three squares

Dear Dr Math:

What numbers cannot be expressed as the sum of three squares? 

I will need the theory behind the answer. Many thanks.

Date: 05/18/98 at 15:14:56
From: Doctor Rob
Subject: Re: Sum of three squares

Theory:    Integer squares are congruent to 0, 1, or 4 (mod 8).

Theorem:   Any integer congruent to 7 (mod 8) cannot be represented as 
           the sum of three squares.

Proof:     Suppose N = 7 (mod 8), and N = x^2 + y^2 + z^2. Then, 
           reducing modulo 8, you must have 7 as a sum of three 
           numbers, each from the set {0,1,4}. Since N is odd, an odd 
           number of the three squares must be odd, and so an odd 
           number of these three numbers must be 1. If all three are 
           1, the sum is 3, not 7. If one of them is 1, then the other 
           two must be even and add to 6, which is 2 mod 4, while the 
           numbers are 0 mod 4. This is impossible.


Corollary: Any integer N of the form 4^m * (8*k + 7) cannot be written 
           as the sum of three squares.

Proof:     If m > 1, and N = x^2 + y^2 + z^2 = 0 (mod 4), then each of 
           x, y, and z must be even. Then:

              N/4 = (x/2)^2 + (y/2)^2 + (z/2)^2 

           is also representable as the sum of three squares. 
           N/4 = 4^(m-1)*(8*k+7) is thus representable as the sum of 
           three squares. Continue the reduction of the exponent of 4 
           until it is zero, and then you will have concluded that 
           8*k + 7 is representable, which is false by the theorem.

The proof that these are the only integers not so representable was
given by Legendre, in _Essai sur la theorie des nombres_ (1798), pp. 
202, 398-9, and Gauss, _Disquistiones Arithmeticae_ (1801), Section 
291. It depends on the theory of ternary quadratic forms. I do not 
know the details, but it is very complicated. There is an alternative 
proof that is more elementary, longer, and also very complicated, 
which is due to Liouville and Uspensky. This appears in Uspensky and 
Heaslet, _Elementary Number Theory_ (1939), pp. 465-474.

-Doctor Rob,  The Math Forum
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