Tamref's Last Theorem

Theorem
The Diophantine equation:
 * $n^x + n^y = n^z$

has exactly one form of solutions in integers:


 * $2^x + 2^x = 2^{x + 1}$

Proof
Since $n^z = n^x + n^y > n^x$ and $n^y$, $z > x,y$.

assume that $x \le y < z$.

Since both $n^{y - x}$ and $n^{z - y} - 1$ are positive integers, both are equal to $1$.

This gives $y = x$ and $n^{z - y} = 2$,

which gives the integer solution $n = 2$, $z - y = 1$.

Thus the solutions are $\tuple {n, x, y, z} = \tuple {2, x, x, x + 1}, x \in \N$.