Number Theory: The Secret Life of Integers
If you have ever entered a credit card number on a website, sent a private message, or logged into a service that uses HTTPS, you have used number theory. The difference is that the computer knew it was using number theory, and you did not.
This guide fixes that. We are not going to prove theorems for the sake of proving theorems. We are going to start from a simple question - "can every number be built from smaller numbers?" - and follow it to the doorstep of modern cryptography. By the end, you will understand why multiplying two large primes is easy, but undoing it is nearly impossible, and why that single fact keeps your data safe.
This is the seventh guide in the Mathematics track. It assumes the number families from Numbers & Number Systems and the modular arithmetic from Phase 3 of that guide. If you can do long division and understand what a remainder is, you are ready.
How to read this
- Here for the "how does my browser stay safe" answer? Start with Phase 1 - primes as unique fingerprints.
- Want the full story? Read in order - the clock math in Phase 2 sets up the cryptography in Phase 3.
The phases
- Primes and the Building Blocks of Numbers - prime factorization as a unique fingerprint, and why some numbers are the atoms of arithmetic.
- The Clock Math You Already Know - modular arithmetic extended: remainders, modular exponentiation, and why the same trick that tells you what time it is also secures your data.
- How the Internet Stays Secret - RSA encryption explained as "two large primes multiplied together are easy to compute but nearly impossible to reverse," with the builder's guide to hashing and checksums.
This builds on Numbers & Number Systems (integers, primes, modular arithmetic) and pairs with Counting & Combinatorics (the pigeonhole principle). It is the discrete math backbone of modern computing.