New lower bounds for Romanoff's constant

Romanoff proved in 1934 that the integers of the form p + 2^a, with p prime and a ≥ 1, have positive lower density d. We prove that d ≥ 0.110825 and, conditionally on an upper bound for prime pairs that Johnston and Trudgian derive from Lichtman’s work on primes in arithmetic progressions, that d ≥ 0.119178; the best lower bound so far was 0.10788. The proof refines the method of Elsholtz and Schlage-Puchta, which bounds the second moment of the number of representations in each residue class modulo a fixed integer. Both bounds are certified by computer and also proved in the Lean proof assistant, with the analytic inputs as hypotheses.

Version 11 revises the text of the paper: the abstract and the introduction are rewritten, and the computation and the formal proofs are now Appendices A and B. The theorems and their proofs are those of version 10.

The programs, the data and the Lean files are archived at doi:10.5281/zenodo.23715955 and available at https://github.com/tejasvi/romanoff-constant/tree/v11 (commit 92b5fbd2aa23).