Prove 3-Square Prime Sum Equals One of Primes = 3

[tarheelborn]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

I really have no idea where to start this one.

 

[Mark44]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

I really have no idea where to start this one.

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

 

[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.

 

[Dick]

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

 

[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

 

[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?

 

[Dick]

Sure, unless one of the primes is 3.