NUMBER
THEORY
The study of the integers. While it was once considered the purest, most abstract branch of mathematics, it has become the foundational science behind modern cryptography and digital security.
The Fundamental Theorem
The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either prime itself or can be uniquely factored into a product of prime numbers.
For example, the number 60 can be broken down into $2^2 \times 3 \times 5$. No other combination of primes will ever multiply to exactly 60.
RSA & Cryptography
Why do primes matter outside of pure theory? Because factoring large numbers is computationally asymmetric.
If you take two massive prime numbers ($p$ and $q$) and multiply them together to get $N$, a computer can do that math in milliseconds. But if you give a computer $N$ and ask it to find the original $p$ and $q$, it could take a supercomputer millions of years. This "trapdoor function" secures the entire internet.
Atomic Factorization
Fundamental Theorem Lab
Theoretical Branches
Select a sub-discipline to initialize the interactive laboratory.
Primes & Divisibility
The Sieve of Eratosthenes, prime counting function, and the Riemann Hypothesis.
Modular Arithmetic
Clock math, congruences, and Fermat's Little Theorem.
Cryptographic Systems
RSA encryption, public/private keys, and trapdoor functions.
Diophantine Equations
Polynomial equations where only integer solutions are accepted.