Number theory Series of 5 56 min
Primitive Sets
Why the primes outweigh every other set of integers where none divides another: the shape of Lichtman's 2022 proof, starting from divisibility.
What you need Substitution in integrals, the chain rule, and the derivative of arctan.
Start with Vol. I 2025
- I No One Divides Another Divisibility, unique factorization and primitive sets from scratch, and why counting and natural density cannot say which primitive set is largest.
- II The Erdős Sum Builds the Erdős sum, which weighs each integer a by 1/(a log a), shows why that weight sits at the edge of convergence, and why the primes score 1.6366.
- III Territories and the e^γ Bound How disjoint territories, a density budget of 1 and Mertens' theorem prove Erdős's 1935 bound for primitive sets, sharpened by Lichtman and Pomerance to e^γ.
- IV Slots and the √v Packing Splitting the Erdős primitive set problem into one slot per prime, measuring a composite's distance from prime, and Lichtman's √v packing bound.
- V The Quarter-Turn How the composite discount and the √v packing bound integrate to π/4, proving every odd prime Erdős-strong, and how Lichtman handles the prime 2.