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In the previous post we discussed the number of primes less than a given number and derived some very poor estimates for this quantity. In this post, using no extra technical machinery whatsoever, we derive a slightly better estimate.

Previously, we used Euclid’s proof of the infinitude of the primes as inspiration for a way of arriving at an upper bound for the number of primes less than a given number. Here, we will come up with a slightly cleverer, more quantitative proof of Euclid’s Theorem and, as before, our estimate will grow out of the proof of Euclid’s Theorem, the improvement in the bound being testament to the fact that the proof is more illuminating.

If we are prepared to accept that every number greater than one has a prime divisor, then Euclidâ€™s famous proof of the infinitude of prime numbers is one sentence long: Given any prime number p, if I multiply together all the prime numbers less than or equal to p and then add one, I get a number which leaves a remainder of one when divided by any of the prime numbers less than or equal to p and whence has a prime divisor greater than p. I try to explain why every number greater than one has a prime divisor in this post.

Today I would like to think a little bit more about prime numbers. Specifically, we will spend some time thinking about the number of prime numbers less than a given a number. We will start by seeing if we can get any quantitative information out of Euclid’s proof itself, before moving on to cleverer ways of achieving this.