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. ?
How many positive integer solutions to $a^x+b^x+c^x=abc$?
6
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abstract-algebra
number-theory
diophantine-equations
exponentiation
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0Have you tried AM-GM to bound the solutions? – 2011-09-01