Number-Theory

Building on the prior lesson of the Division Algorithm and Modular Arithmetic, we move onto Prime Factorization and Euclid’s Algorithm. These are not only extremely interesting facets of math and number theory, but one of the most utilized pieces of math in the modern digital world. Prime Factorization is fundamental to almost all encryption, and it is by the sheer scale of the math that allows it to be so — for even simple encryption breaking, a conventional computer could take over a hundred years running full bore before it could crack some types of encryption. This is also what makes things like Quantum Computers so terrifying — they have the capacity to break prime factorization, and thus, our entire modern digital security infrastructure. ...

More often than not, the first mathematical object we are exposed to is Integers. Integers are whole numbers, things like 1, 2, 3. These are simple, and relatively profound in their ability to represent quantities and otherwise unrelated topics. In fact, we have studied integers, known as Number Theory, for thousands of years with no real discernable purpose — until recently. That is to say, not all science comes with some clear endgoal, but it is the search in of itself that provides us with dividends. Number Theory is one such case — where we have built out a system that allows us to encrypt large amounts of data using nothing more than prime factorization for exceptionally large numbers — something that up until today, was almost entirely pointless. ...