Distribution of Primes

Number Theory · Axiom Academy

Exploring how prime numbers are distributed among the integers and the patterns that emerge as we count them 1. The Prime Counting Function For example, the primes up to 10 are 2, 3, 5, 7, so π(10) = 4. As we look at larger numbers, we can tabulate the prime counting function: The animation below shows how π(x) grows as we count primes among the first integers. Although there are infinitely many primes, they become increasingly sparse as we look at larger numbers. The density of primes near x can be measured by the ratio π(x)/x, which represents the proportion of numbers up to x that are prime. The Prime Number Theorem (proved in 1896) gives us a precise asymptotic formula for π(x), showing that primes thin out at a logarithmic rate. Before the Prime Number Theorem was proved, Pafnuty Chebyshev (1850s) established that π(x) grows roughly like x/ln(x), proving the existence of positive constants c₁ and c₂ such that: This was a major breakthrough, showing that while we couldn't yet determine π(x) exactly, we could trap it between bounds that grow at the same rate. The animation illustrates how π(x) stays within these bounds. 4. Prime Gaps Can Be Arbitrarily Large A prime gap is the difference between consecutive primes. While primes never stop appearing, we can find arbitrarily long stretches of consecutive composite numbers. For example, with n = 5, the numbers 5! + 2 = 122, 123, 124, 125, 126 are all composite. Yet despite these large gaps, the primes continue infinitely!

This is the written version of the interactive lesson above. See the full Number Theory course.