Square Root of Prime is Irrational

 

    

Theorem

The square root of any prime number is irrational.

Proof

Let p be prime.
Suppose that p √ is rational.
Then there exist natural numbers m and n such that:

				p √	=		m n
		⟹		p	=		m 2 n 2
		⟹		n 2 p	=		m 2

Any prime in the prime factorizations of n 2 and m 2 must occur an even number of times because they are squares.
Thus, p must occur in the prime factorization of n 2 p an odd number of times.
Therefore, p occurs as a factor of m 2 an odd number of times, a contradiction.
So p √ must be irrational.

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