Diophantine Equation for Odd Numbers

Question

Dealing with Diophantine equation I saw the following to be true and could arrive at a proof. Has this been dealt with earlier any where ? $$x^p+y^q=z^r $$, where $$x, y, p, q, z, r$$ are all natural numbers. It can be shown that the above statement is never true when $p $ and $q$ are even integers and $x$ and $y$ are odd integers and $r > 1$

Answer 1

Without that requirement, there are solutions such as $$ \eqalign{0^2 + 1^3 &= 1^2\cr 1^2 + 2^3 &= 3^2\cr 3^2 + 3^3 &= 6^2\cr 6^2 + 4^3 &= 10^2\cr 10^2 + 5^3 &= 15^2\cr 15^2 + 6^3 &= 21^2\cr 21^2 + 7^3 &= 28^2\cr 28^2 + 8^3 &= 36^2\cr 36^2 + 9^3 &= 45^2\cr 45^2 + 10^3 &= 55^2\cr \ldots&\cr }$$