- #1

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## 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.

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- Thread starter tarheelborn
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- #1

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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.

I really have no idea where to start this one.

- #2

Mark44

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## 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

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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

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

- #5

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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

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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

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Sure, unless one of the primes is 3.

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