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 a1 < a2 < … and b1 < b2 < … such that ai + bj is prime for all i < 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.
This video is part of the Number Theory Web Seminar series.
