Prime numbers – Peter’s Blog

on November 24, 2015

Prime numbers fascinate me. They are essentially the building blocks for multiplication. Recall that a positive integer ${n \geq 2}$ is prime if the only positive integers dividing into $n$ are none other than 1 and itself. A well-known property of the integers is that each positive integer can be uniquely written as a product of prime numbers.

There are infinitely many prime numbers. In my opinion, the proof of this result (due to Euclid) is one of the most elegant mathematical proofs. For those who have not seen it, here it is:

Theorem: There are infinitely many prime numbers.

Proof: Suppose on the contrary there are finitely many prime numbers, say $p_1, p_2, \ldots , p_n$ . Form the number $n = p_1 p_2 \cdots p_n + 1$ . Since $n$ is larger than each $p_i$ , it cannot be prime. Thus, it must be divisible by one of the primes, say $p_i$ . But this is impossible since $n$ leaves a remainder of 1 on division by $p_i$ . We have thus arrived at a contradiction so that the original assumption of finitely many prime numbers is false. Thus there are infinitely many prime numbers. $\square$

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