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 $$\color{red}{p^2 /q^2 = 4}$$ then $$p^2 = 4q^2$$, $$\color{blue} {\text{with } p \text{ has a factor of 4}}$$. Let $$p = 4m$$, then $$\color{green}{(4m)^2 = 4q^2 } \rightarrow 16m^2 = 4q^2, 4m^2 = q^2$$, then $$\color{orange}{q \text{ also has a factor of 4}}$$. 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 $$\color{red}{red}$$, "2" represents $$\color{blue}{blue}$$, "3" represents $$\color{green}{green}$$, "4" represents $$\color{orange}{orange}$$.

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