Root 4 is irrational!

Let us prove that $\sqrt 4$ is irrational:

Let us assume that $\sqrt 4$ is rational, then it can be expressed in the form of $\frac pq$ where $p,q$ have no factors in common, thus $\frac pq = \sqrt 4$ . or $\color{#D61F06}{p^2 /q^2 = 4}$ then $p^2 = 4q^2$ , $\color{#3D99F6} {\text{with } p \text{ has a factor of 4}}$ . Let $p = 4m$ , then $\color{#20A900}{(4m)^2 = 4q^2 } \rightarrow 16m^2 = 4q^2, 4m^2 = q^2$ , then $\color{#EC7300}{q \text{ also has a factor of 4}}$ . But this contradicts the fact that $p,q$ are coprimes. Thus the assumption that $\sqrt 4$ is rational is wrong. Hence, $\sqrt 4$ is also irrational.

This concludes that $\sqrt 4 = 2$ is an irrational number.

Which of these above colored equations are not necessarily correct?

"1" represents $\color{#D61F06}{red}$ , "2" represents $\color{#3D99F6}{blue}$ , "3" represents $\color{#20A900}{green}$ , "4" represents $\color{#EC7300}{orange}$ .