My conjecture: There exists no natural number $x$ for which the following statement holds: The string $x$ occurs $x$ times in the number $x^x$.
For example, pretend that $11921192^{11921192}$ contains (when written in base 10) the substring: $\ldots 341192119245 \ldots$ which contains "11921192". If there are 11921191 other occurences of this substring, then this number satisfies my conditions.
I've written a Python program to brute force the numbers:
for x in range(1,100000000):
if str(x**x).count(str(x)) == x:
print x
but it did not find such a number <100 millions. I decided to take a mathematical approach and did some arithmetic in other bases. I do not know what should I try next. I'd like some advice on how to tackle this problem. Thanks in advance.
If $x$ is a power of $10$, then $x^x$ is also a power of $10$ and therefore has only one copy of $x$. This gives $x=1$ as a solution.
If $x$ is not a power of $10$, then $x$ has $t=\lceil \log_{10}x\rceil $ digits (here $\lceil \cdot \rceil$ denotes the ceiling function), while the number $x^x$ has $$\lceil \log_{10}x^x\rceil=\lceil x \log_{10}x\rceil\le \lceil xt\rceil=xt$$
Hence if $x^x$ has $x$ copies of the string '$x$', then it contains nothing else, i.e. $x^x$ is just $x$ copies of that string, concatenated. But then $$x^x=x(1+10^t+10^{2t}+\cdots+10^{xt})=x\frac{10^{xt+1}-1}{10-1}$$ which rearranges as $$9x^{x-1}=10^{xt+1}-1$$
For large $x$, the LHS is less than $9x^x$, while the RHS is greater than $10 x^x$, so these are never equal. Checking small $x$ (say up to 10) shows only $x=1$ as a solution.