Sum of 3 squares

  • #1
123
0

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.
 

Answers and Replies

  • #2
35,052
6,791

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.
 
  • #3
123
0
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
Dick
Science Advisor
Homework Helper
26,260
619
If p is a prime different from 3, what is p^2 mod 3?
 
  • #5
JDW
1
0
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
123
0
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
Dick
Science Advisor
Homework Helper
26,260
619
Sure, unless one of the primes is 3.
 

Related Threads on Sum of 3 squares

  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
6
Views
3K
  • Last Post
Replies
4
Views
4K
  • Last Post
Replies
1
Views
4K
  • Last Post
Replies
4
Views
10K
  • Last Post
Replies
5
Views
1K
Replies
1
Views
4K
Top