This is Exercises 11 from Section 7.3 of the book "Abstract Algebra: Theory and Applications" (aata-20180801) by Thomas W. Judson.
Find integers $n$, $E$, $X$ such that $$ X^{E} \equiv X \;(\text{mod}\; n). $$ Is this a potential problem in the RSA cryptosystem?
What does the "potential problem" mean in this problem? Is it related to the Iterated Encryption Attack described in this article?
Assume that you find integers $n$, $E$, $X$ such that $$ X^{E} \equiv X \;(\text{mod}\; n). $$ Then;
$$ X^{E} -X = k \cdot n \quad\text{ for some } k\in \mathbb{Z}.$$
Take out $X$
$$ X ( X^{E-1} -1) = k \cdot n \quad (= k \cdot p \cdot q)$$
Now, calculate $\gcd(X,n)$ and $\gcd(X^{E-1} -1,n)$.
If you are lucky, you might find $p$ and $q$.