2005/10/06 by Bryna Kra · 2 citations
Mathematics · #Analytic Number Theory Research #Limits and Structures in Graph Theory #Algebraic Geometry and Number Theory #Conjecture #Ergodic theory #Annotation #Prime (order theory) #Mathematics #Semantics (computer science) #Type (biology) #Algorithm #Computer science #Discrete mathematics #Arithmetic #Artificial intelligence #Combinatorics #Pure mathematics #Programming language
paper · pdf · doi:10.1090/s0273-0979-05-01086-4
openalex publication_date 2005/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
A long-standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only prog- ress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , there exist infinitely many arithmetic progressions of length <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory.