The contradiction method used to prove that the square root of a prime is irrational The contradiction method given in certain books to prove that sqare root of a prime is irrational also shows that sqare root of $4$ is irrational, so how is it acceptable?
e.g. Suppose $\sqrt{4}$ is rational,
$$\begin{align}
       \sqrt{4} &=p/q        \qquad\text{where pand q are coprimes} \\
        4 &=p^2/q^2\\
      4q^2&=p^2       \tag{1} \\
         4&\mid p^2\\
         4&\mid p\\
     \text {let }p&=4m  \qquad\text{for some natural no. m}  \\
       p^2&=16m^2\\
      4q^2&=16m^2   \qquad\text{(from (1) )}\\
       q^2&=4m^2\\
         4& \mid q^2\\
         4&\mid q
\end{align}
$$
but this contradicts our assumption that $p$ and $q$ are coprime since they have a common factor $p$. Hence  $\sqrt{4}$ is not rational. But we know that it is a rational. Why?

Let $p$ be prime and suppose that $\sqrt{p}$ is rational. Then $\sqrt{p}=\frac{r}{s}$ for some integers $r$ and 
$s$ with $s$ not zero. Then $s^2 p=r^2=q_1 \ldots q_n$ for some primes $q_i$ by the fundamental theorem of 
arithmetic. Note that $r^2$ and $s^2$ have an even number of prime factors by expanding squares. But then 
$s^2 p$ has an odd number of prime factors. This is a contradiction. Then $\sqrt{p}$ is irrational.
