Number Theory

Essential formulas and concepts in number theory

Fermat's Little Theorem

\(a^{p-1} \equiv 1 \pmod p\)

Variables:

  • \(p is a prime number, a is an integer not divisible by p.\)

Description/Usage:

In modular arithmetic, this theorem states that if \(p\) is prime, then \(a^{p-1}\) leaves a remainder of \(1\) upon division by \(p\). It’s fundamental for simplifying exponents mod \(p\).

Mathematical concept visualization

Euler's Totient (Phi) Formula

\(\varphi(n) = n \displaystyle\prod_{p \mid n}\Big(1 - \frac{1}{p}\Big)\)

Variables:

  • \(n is a positive integer, and the product is over each prime p dividing n.\)

Description/Usage:

Mathematical concept visualization