In trying to write an alternate and simple proof that at least one leg of a right triangle is a multiple of 4 using Dickson's method of generating triples, I came across quite an interesting observation that ifall the sides of a rectangle is odd, then it's diagonal is irrational. For example, consider a rectangle of length 5 units and breadth 3 units. It's diagonal,by the Pythagorean theorem is 34^0.5, which is irrational. This is the same for lots of rectangles. Is there a general proof for this, or can this whole thing be disproved?
[Math] Prove that if a rectangle’s sides are all odd, then it’s diagonal is irrational
pythagorean triples
Best Answer
$$(2a+1)^2+(2b+1)^2=4a^2+4a+4b^2+4b+2=2(2c+1).$$
This number cannot be a perfect square as its prime factorization includes $2^1$.