# Sum of 3 squares

1. Apr 22, 2010

### tarheelborn

1. The problem statement, all variables and given/known data

Prove that if a prime number is a sum of three squares of different primes, then one of the primes must be equal to 3.

2. Relevant equations

3. The attempt at a solution

I really have no idea where to start this one.

2. Apr 22, 2010

### Staff: Mentor

Start with an equation that represents the given part of what you're trying to prove.

3. Apr 23, 2010

### tarheelborn

So something like:

Let p, q, r, and s be prime. Then if s = p^2 + q^2 + r^2, either p, q, or r must = 3.

The only theorem I have on 3 squares is that N >=1 is a sum of three squares if and only if N <> 4^n(8m+7), for some m, n >= 0.

4. Apr 23, 2010

### Dick

If p is a prime different from 3, what is p^2 mod 3?

5. Apr 23, 2010

### JDW

A couple of hints:

Try writing your primes as p = 3k + r, r = 0, 1, 2. (Note if k != 1, r cannot be zero, then p isn't prime)

Consider values mod 3

6. Apr 26, 2010

### tarheelborn

So p would have to be 1(mod 3) ==> a^2+b^2+c^2==0(mod 3) ==> 3|p which is a contradiction, right?

7. Apr 26, 2010

### Dick

Sure, unless one of the primes is 3.