TITLE: How many positive integer solutions to $a^x+b^x+c^x=abc$?
QUESTION [6 upvotes]: How many positive integer solutions are there to $a^{x}+b^{x}+c^{x}=abc$? (e.g the solution $x=1$, $a=1$, $b=2$, $c=3$). Are there any solutions with $\gcd(a,b,c)=1$? Any solutions to $a^{x}+b^{x}+c^{x}+d^{x}=abcd$ etc. ?

REPLY [2 votes]: The $x=1$ problem, $$a_1+a_2+\cdots+a_k=a_1a_2\cdots a_k$$ is discussed at some length at D24 in Guy's Unsolved Problems In Number Theory. There is always the solution $a_1=2$, $a_2=k$, $a_3=a_4=\cdots=a_k=1$. Various people have shown that $k=2,3,4,6,24,114,174$, and $444$ are the only $k\le1440000$ for which there is exactly one solution. See the book for more questions, references, etc.