# Pythagorean Triples

• Dec 8th 2008, 07:05 PM
mndi1105
Pythagorean Triples
Let x,y,z be a primitive Pythagorean Triple with y even.

a)Prove that exactly one of x and y is divisible by 3. (Hint: Proof by contradiction)

b)Prove that exactly one of x and y is divisible by 4.
• Dec 8th 2008, 10:18 PM
CaptainBlack
Quote:

Originally Posted by mndi1105
Let x,y,z be a primitive Pythagorean Triple with y even.

a)Prove that exactly one of x and y is divisible by 3. (Hint: Proof by contradiction)

The square of a number not divisible by $3$ is congruent to $1$ modulo $3$ (prove this). And no square is congurent to $2$ modulo $3$ (prove this also)

Suppose $x$ and $y$ are not divisible by $3$, then the left hand side of:

$
x^2+y^2=z^2
$

is congruent to $2$ modulo $3$ while the right hand side cannot be congurent to $2$ modulo $3$, a contradiction, hence at least one of $x$ and $y$ is divisible by $3$.

If both $x$ and $y$ are divisible $3$ then the left hand side is divisible by $3$ but the right hand side cannot be divisivle by $3$ because $\{x,y,z\}$ is a primitive Pythagorean triple and so $x,y,x$ share no common factor. This is a contradiction, hence both $x$ and $y$ cannot be divisible by $3$.

Hence exactly one of $x,y$ is divisible by $3$.

(note we have not used the fact that $y$ is even)

CB