Green–Tao theorem

The Green–Tao theorem states that the prime numbers contain arithmetic progressions of every finite length, despite their density tending to zero.

Transference principle: A method for transferring results about dense sets to sparse sets that behave sufficiently like random sets. It lets a dense arithmetic-progression theorem yield one for the sparse primes.
Arithmetic progression: A sequence of numbers in which the difference between consecutive terms is constant. The theorem concerns progressions whose terms are all prime.
Ben Green: A British mathematician whose work includes additive combinatorics and analytic number theory. He co-authored the proof and developed key tools for transferring dense results to primes.
Polynomial progressions in the primes: Patterns among primes whose terms are values of specified polynomials and satisfy prescribed progression relations. The theorem's methods inspired extensions toward more varied polynomial patterns in primes.
Quantitative Green–Tao theorem: A strengthened form of Green–Tao giving bounds on where progressions of a specified length can be found. The original qualitative guarantee leaves the size of the first such progression poorly controlled.
Pseudorandomness: A resemblance to randomness measured through specified statistical tests rather than complete probabilistic independence. The prime weights satisfy structured tests that make the transference step possible.
Prime number: A natural number greater than one whose only positive divisors are one and itself. These are the terms from which the theorem's progressions are formed.
Terence Tao: An Australian-American mathematician whose research spans analysis, number theory, and combinatorics. He co-authored the theorem and helped develop its transference framework.
Primes in arithmetic progression: A sequence of primes with a fixed common difference, such as 5, 11, 17, and 23. The theorem allows a different common difference for each progression, unlike a fixed-difference prime sequence.
Least prime in an arithmetic progression: The smallest prime that begins a prime arithmetic progression of a specified length. Bounding this quantity would make the theorem's existence guarantee more concrete.

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