# Infinite Partial Sumsets in the Primes

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 less than a_2 less than ... and b_1 less than b_2 less than ... such that a_i + b_j is prime for all i less than 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.

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### Affiliation

University of California, Los Angeles; Member, School of Mathematics