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The study of whole numbers is one of the oldest lines of inquiry in mathematics which is still thriving today. After all, it is possible in pure mathematics for the relevance of work done thousands of years ago to be undiminished today. I want to talk today about what is possibly the prototypical example of such a piece of mathematics. I am referring to Euclid’s proof of the infinitude of prime numbers.
A divisor of a number n is another, smaller number, for which there is no remainder when n is divided by it. For example, three is a divisor of six, seven is a divisor of seven and thirty-nine is a divisor of one hundred and seventeen. A prime number is a number with exactly two divisors. Let’s think about what it means to have exactly two divisors. Firstly, we observe that every whole number has at least one divisor, namely the number itself. Secondly, every whole number except for ‘1’ has at least two divisors: The number itself and the number ‘1’. So, with the exception of the number ‘1’, being prime expresses the quality of having the fewest possible divisors. Why must there must be infinitely many such numbers?