In This Lesson Prime Numbers Fundamental Theorem of Arithmetic 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 GCD & LCM The Sieve of Eratosthenes 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 Distribution of Primes 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.
Fundamental Theorem of Arithmetic
Every integer n > 1 has a unique prime factorization:
n = p₁^a₁ · p₂^a₂ · … · pₖ^aₖ
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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).