A, B, C are 3 distinct primes.
Find the smallest composite number Y that satisfies the relation:
$ Y = A ^ C + B ^ B + C ^ A $
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The goal is to minimize, so you'll need to obviously use the primes
$2, 3, 5$
The $5$ can't be in the middle as $5^5$ is large, so it must be on either side. $(2^5 + 3^3) < (3^5 + 2^2)$, so your equation is $2^5 + 3^3 + 5^2$.
$(A, B, C) = (2, 3, 5)$ and $Y = 84$
I totally agree with the answer given by @Aranlyde.
But you don't explicitly specify that $A\not=B\not=C$.
So, I would do something like $A=B=C=2$.
Then, the result would be:
In any way, this is the minimum.