Terence Tao: Infinite Partial Sumsets in the Primes (NTWS 160)
Channel: Number Theory Web Seminar
Date: 2023-04-16

Abstract: It is an open question as to whether the prime numbers contain the sum A+B of two infinite sets of natural numbers A, B (although results of this type are known assuming the Hardy-Littlewood prime tuples conjecture).  Using the Maynard sieve and the Bergelson intersectivity lemma, we show the weaker result that there exist two infinite sequences a_1 \textless a_2 \textless ... and b_1 \textless b_2 \textless ... such that a_i + b_j is prime for all i\textless j.  Equivalently, the primes are not "translation-finite" in the sense of Ruppert.  As an application of these methods we show that the orbit closure of the primes is uncountable.


Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date:
Thursday, February 23, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 24, 2023 (12am CST, 3am AEDT, 5am NZDT)