Tamref's Last Theorem

Theorem

$n^x + n^y = n^z$

has exactly one form of solutions in integers:

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

for all $x \in \Z$.

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

Without loss of generality assume that $x \le y < z$.

$\ds n^x + n^y$

$=$

$\ds n^z$

$\ds \leadsto \ \ $

$\ds 1 + n^{y - x}$

$=$

$\ds n^{z - x}$

Divide both sides by $n^x$

$\ds \leadsto \ \ $

$\ds 1$

$=$

$\ds n^{y - x} \paren {n^{z - y} - 1}$

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:

$\tuple {n, x, y, z} = \tuple {2, x, x, x + 1}, x \in \Z$

$\blacksquare$

Tamref's Last Theorem appears as tamreF's last theorem on the Math Fun Facts page of Harvey Mudd College.

The name is an obvious reversal of the surname of Pierre de Fermat, in homage to the famous Fermat's Last Theorem.