All assuming  :;Lemma 1:   :The following are all equivalent: :#GCD(a', b, c) = 1 :#GCD(a, b) = 1 :#GCD(b, c) = 1 :#GCD(c, a) = 1 :;Proof: :2, 3 and 4 obviously imply 1. :To show that 1 implies 2, assume that 2 fails, and h = GCD(a, b) ≠ 1, then: :: h divides a and b. :: h2 divides a2 and b2. :: h2 divides c2 = a2 + b2. :: h divides c. :: h divides GCD(a', b, c). :: QED :Careful study shows the we have proved that {{nowrap|1=GCD(a', b, c) = GCD(a, b) = GCD(b, c) = GCD(c, a)}}. :;Lemma 2: :The square of an even number is congruent to 0 modulo 4.  The square of an odd number is congruent to 1 modulo 4. :;Proof: :: :: ::QED. :;Theorem ::If (a,b,c) is a PPT then a and b are of opposite parity and c is odd. :;Proof: ::If a and b are both even, then 2 | GCD(a, b) , so (a,b,c) is not a PPT. ::If a and b are both odd, then a2 + b2 is congruent to 2 modulo 4, a contradiction. ::If a and b are of opposite parity, then a2 + b2 is congruent to 1 modulo 4, and c'' is odd. : —     :QED.  