The atoms of arithmetic — primes are the building blocks of all integers.
A prime p > 1 has exactly two divisors: 1 and itself. The first primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …
Primes are the foundation of modern cryptography and connect deeply to factoring in algebra.
360 ÷ 2 = 180 → 180 ÷ 2 = 90 → 90 ÷ 2 = 45 → 45 ÷ 3 = 15 → 15 ÷ 3 = 5 → 5 ÷ 5 = 1
Divisors of 360: (3+1)(2+1)(1+1) = 24 divisors
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.
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 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).