TITLE: Is the "cyclotomic diagonalization" always squarefree?
QUESTION [22 upvotes]: For every integer $n\ge 2$ , define $$f(n):=\Phi_n(n)$$ where $\Phi(n)$ is the $n$ th cyclotomic polynomial. This can be considered as the "cyclotomic diagonalization"

Prove or disprove the conjecture that $f(n)$ is squarefree for every integer $n\ge 2$

The partial factorizations I determined with PARI/GP (usually, the factors upto $12-13$ digits are found) gave no exponent greater than $1$ upto $n=200$. A rigorous test upto $n=46$ revealed that upto this limit there are only squarefree values.
If it cannot be proven, a good heuristic would also be welcome.

REPLY [11 votes]: Checking the pairs $n,p\le 10^6$ , I found one counterexample , namely :
$$283411^2\mid \Phi_{28341}(28341)$$