If a triple of positive integers $(a,b,c)$ satisfies $a^2+b^2=c^2$, it is called a Pythagorean triple. No triple $(a,b,c)$ satisfies $a^e+b^e=c^e$ when $e\ge 3$ (Fermat's Last Theorem). However, if the exponents of the left-hand side and right-hand side differ, this is not true. For example, $3^3+6^3=3^5$.

Let $a,b,c,e,f$ be all positive integers, $0<a<b$, $e\ge 2$, $f\ge 3$ and $c^f\le N$. Let $F(N)$ be the number of $(a,b,c,e,f)$ such that $a^e+b^e=c^f$. You are given $F({10}^3)=7$, $F({10}^5)=53$ and $F({10}^7)=287$.

Find $F({10}^{18})$.