Hello! This is my first ever blog post. I’m finding that the only way to get better at writing and expressing my thoughts is by, well, writing and expressing my thoughts.

MAY 1, 2023

Recently, I’ve taken to (re)teaching myself mathematics. At the writing of this post I am 26 and so a natural question people ask is why? There could be a longer post answering this question but in short there a few reasons driving me:

1. I’m a software engineer and mathematical reasoning skills seem to be table stakes for truly notable success and contributions in technical fields.
2. I want to be able to better model/quantify the risk associated with various trades/investments I make.
3. I’ve grown to appreciate and become more interested in math as I’ve gotten older.

To that end, one of the first books I’m going through is on mathematical proofs, How To Prove It by Daniel J. Velleman. In one of the first chapters he introduces Euclid’s theorem, a fundamental theorem in number theory that states there are infinitely many prime numbers. I thought it’d be fun (because what could be more fun?) to walk through this proof and attempt to make it more digestible. Selfishly, I’m doing this to cement the knowledge in my own mind, but as part of the intent of these blogs to convey ideas, this seems like good practice at breaking down ideas and attempting to effectively communicate them!

Euclid’s theorem

In his work Elements, Euclid (Greek mathematician) asserts the following:

Prime numbers are more than any assigned multitude of prime numbers.

In layman’s terms, he’s asserting that there are infinitely many prime numbers. As a refresher, an integer greater than 1 is either:

• prime if it cannot be expressed as a product of two smaller integers (excluding 1)
• composite if can be expressed as a product of two smaller integers (excluding 1)
  • one feature of composite integers is that they can be written as products of prime numbers

Without a proof, Euclid’s statement is merely conjecture, so let’s break down the proof he provides.

Consider:

n can be any positive integer greater than 1, so f can be any finite list of prime numbers. Now suppose we take tall the elements of f and produce a product, which we call P

Now let q=P+1 , where q is equal to the product of all of the primes in list f, otherwise known as P, plus 1. Earlier I said an integer greater than 1 is either prime or composite. Euclid’s proof makes two statements about this new value q :