A prime p > 1 has exactly two divisors: 1 and itself. The first primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …
2 is the only even prime — every even number > 2 is divisible by 2. To test if n is prime, check divisibility by primes up to √n (why? if n = a·b, one factor must be ≤ √n).
GCD(a, b) = product of common primes with min exponents
LCM(a, b) = product of all primes with max exponents
a · b = GCD(a, b) · LCM(a, b)
Euclidean Algorithm computes GCD efficiently: GCD(a, b) = GCD(b, a mod b). This is one of the oldest algorithms — and it's essential in modular arithmetic for finding modular inverses.
The Sieve of Eratosthenes
To find all primes ≤ n: start with 2, mark all multiples of 2, next unmarked (3), mark all multiples of 3, continue to √n. The remaining unmarked numbers are prime. Complexity: O(n log log n).
The Prime Number Theorem: π(n) ≈ n/ln(n), where π(n) counts primes ≤ n. This connects primes to logarithmic functions and limits. There are infinitely many primes (Euclid's proof by contradiction is one of the most elegant in mathematics).
Open problems: the Twin Prime Conjecture (infinitely many primes p where p+2 is also prime), Goldbach's Conjecture (every even n > 2 is the sum of two primes), and the Riemann Hypothesis (about the precise distribution of primes).