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Is there some (huge) positive integer $M$ with the following property: for any $z>M$, there exist positive integers $x, y_{1}, y_{2},..., y_{z}$ such that $x^x$ $=$ $y_{1}^{y_{1}}$+ $y_{2}^{y_{2}}$+ ... +$y_{z}^{y_{z}}$ ?

[Please remark that the $y$'s are $\geq$ $1$ and need not to be necessarily distinct.]

As a [rather naive] way to attack this problem [which may (perhaps) be related to some works of Robinson, Matiasevich, M. Davis, and Chao-Ko], I'm thinking about lots of $1$'s, lots of $2$'s, and lots of $(x-1)$'s. Also, let us observe that, if $z$ has this property and $y_{i}$ $=$ $2$ for some $i$, then $z+3$ has the same property, too...

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Have you tried looking at the density of possible $z$'s?

I think the answer might be "no". Here's my heuristics (hoping it's not bogus): the smallest $z$ we can achieve using $x$ is at least $\frac{x^x}{(x-1)^{(x-1)}}\approx ex$, the second smallest would be at least $\frac{x^x-(ex-1)(x-1)^{(x-1)}}{(x-2)^{(x-2)}}\approx e^2x$ and so on.

Let $N$ be a very large integer. we look at the interval $[1,N]$ and see how many such $z$'s are in it. From the above argument there are at most $\approx \frac{N}{e}+\frac{N}{e^2}+\cdots=\frac{N}{e-1}< N$ solutions. This contradicts your conjecture that there are asymptotically $\approx N$ solutions (in terms of values of $z$)

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