The lucky cab

The $LHS$ is a sum of two exponents ${ 17 }^{ \sqrt { x-1729 } }$ and ${ 29 }^{ \sqrt { x-1729 } }$ and is increasing since the bases of the exponents are greater than $1$ .

Combining this with the fact that $\min { \left\{ { 17 }^{ \sqrt { x-1729 } }+{ 29 }^{ \sqrt { x-1729 } } \right\} } =2<17.29$ at $x=1729$ we can see that there is exactly one point of intersection of both functions.