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.
The square of a number not divisible by is congruent to modulo (prove this). And no square is congurent to modulo (prove this also)
Suppose and are not divisible by , then the left hand side of:
is congruent to modulo while the right hand side cannot be congurent to modulo , a contradiction, hence at least one of and is divisible by .
If both and are divisible then the left hand side is divisible by but the right hand side cannot be divisivle by because is a primitive Pythagorean triple and so share no common factor. This is a contradiction, hence both and cannot be divisible by .
Hence exactly one of is divisible by .
(note we have not used the fact that is even)
CB