TITLE: Not quite Fermat's Last Theorem
QUESTION [20 upvotes]: Prove that the equation $n^a + n^b = n^c$, with $a,b,c,n$  positive integers, has infinite solutions if $n=2$, and no solution if $n\ge3$.

REPLY [10 votes]: Wlog $\,a \le b$. Dividing by $n^a$ yields $\,1 + n^{b-a} = n^{c-a}$ $\Rightarrow$ $b=a\ $ (else $\,n\mid1)\,$ $\Rightarrow$ $\, n = 2,\, c = a\!+\!1$.