Given any positive integer $n$ and prime number $p$, I would like to show that one can write $n$ as a sum of positive integers $a+b$ so that $a^ab^b$ is not congruent to $n^n$ mod $p$. One can easily show this is true for $n \ge p$, so one can restrict attention to the case where $n$ is less than $p$.

Experimentally, Mathematica tells me that for all $n$ less than 10,000 one can get away with choosing $a$ to be less than $4$ (ie there is no number in that range with $n^n \equiv (n-1)^{(n-1)} \equiv 4(n-2)^{(n-2)} \equiv 27(n-3)^{(n-3)}$ mod $p$), and in fact the least residues of the products we are interested in tend to have very little overlap, so this is "certainly" true (as much as we can ever say that in mathematics without having a proof), but I don't see a way to prove it. Any thoughts?

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