Combinatorics

number theory studies the integers on their own terms — divisibility, primes, remainders — and for two millennia it was the purest of pure mathematics, prized precisely because it was useless. 𐃏 then in 1977 the RSA cryptosystem turned euclid, fermat and euler into load-bearing internet infrastructure. this page walks the classical results in dependency order and ends at that payoff (Epp, Susanna S., 2019).

graph theory began in 1736 when euler noticed that a walking-tour puzzle about seven bridges in königsberg was really a question about dots and lines — and that the geometry (lengths, positions, the river) was irrelevant. 𐃏 that act of abstraction — keep the incidence structure, discard everything else — is the whole subject. a graph is the minimal mathematical model of “things, some pairs of which are related”, which is why it keeps turning up: networks, molecules, dependencies, state machines, social graphs (Epp, Susanna S., 2019).

discrete mathematics is the mathematics of structures that come in separate, countable pieces — integers, truth values, finite sets, graphs — as opposed to the smooth continuum of analysis. 𐃏 no limits, no derivatives; instead: induction, counting, and exact structure. it is the native mathematics of computation — every data structure, algorithm, database query and cryptographic protocol below this page is discrete mathematics wearing a lab coat (Epp, Susanna S., 2019).