<s>[INST] Divisor of $x^2+x+1$ can be square number? $$1^2+1+1=3$$
$$2^2+2+1=7$$
$$8^2+8+1=73$$
$$10^2+10+1=111=3\cdot37$$
There is no divisor which is square number.
Is it just coincidence? Or can be proved?
*I'm not english user, so my grammer might be wrong
 [/INST] No, for $x=18$ we get $x^2+x+1=343=7^3$.
Here are the first few counterexamples:
$$
\begin{array}{rrl}
x &  x^2+x+1 & \text{factorization}\\
18 & 343 & 7^3 \\
22 & 507 & 3 \cdot 13^2 \\
30 & 931 & 7^2 \cdot 19 \\
67 & 4557 & 3 \cdot 7^2 \cdot 31 \\
68 & 4693 & 13 \cdot 19^2 \\
79 & 6321 & 3 \cdot 7^2 \cdot 43 \\
116 & 13573 & 7^2 \cdot 277 \\
128 & 16513 & 7^2 \cdot 337 \\
146 & 21463 & 13^2 \cdot 127 \\
165 & 27391 & 7^2 \cdot 13 \cdot 43 \\
177 & 31507 & 7^2 \cdot 643 \\
191 & 36673 & 7 \cdot 13^2 \cdot 31 \\
214 & 46011 & 3 \cdot 7^2 \cdot 313 \\
\end{array}
$$
</s>