Prime Numbers & Divisibility

The atoms of arithmetic — primes are the building blocks of all integers.

Prime Numbers

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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).

Primes are the foundation of modern cryptography and connect deeply to factoring in algebra.

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Fundamental Theorem of Arithmetic

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Every integer n > 1 has a unique prime factorization:
n = p₁^a₁ · p₂^a₂ · … · pₖ^aₖ

Example: 360 = 2³ · 3² · 5

360 ÷ 2 = 180 → 180 ÷ 2 = 90 → 90 ÷ 2 = 45 → 45 ÷ 3 = 15 → 15 ÷ 3 = 5 → 5 ÷ 5 = 1

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Divisors of 360: (3+1)(2+1)(1+1) = 24 divisors

GCD & LCM

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).

Distribution of Primes

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).

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