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On the inverse Goldbach problem
Abstract
We establish two quantitative results concerning additive decompositions of the set ℙ of primes. First, we prove that, if A,B⊆ℕ have at least two elements and A+B differs from the primes by only finitely many elements, then √x/(log x)≪ A(x),B(x) ≪ √x. Second, we show that, if A,B,C⊆ℕ have at least two elements each, then ((A+B+C)mathbinbigtriangleupℙ)(x) ≫ √x/((log x)2). An analogous estimate is obtained for sumsets of k≥ 3 sets. The proofs use Montgomery's large sieve, Gallagher's larger sieve, a mean-value argument for a certain multiplicative function, and a submultiplicative sumset inequality.
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BibTeX
@misc{leonetti2026inverse,
title = {{On the inverse Goldbach problem}},
author = {Paolo Leonetti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15758},
eprint = {2610.15758},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15758},
}
RIS
TY - UNPB
TI - On the inverse Goldbach problem
AU - Paolo Leonetti
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15758
UR - https://arxiv.org/abs/2610.15758
AB - We establish two quantitative results concerning additive decompositions of the set ℙ of primes. First, we prove that, if A,B⊆ℕ have at least two elements and A+B differs from the primes by only finitely many elements, then √x/(log x)≪ A(x),B(x) ≪ √x. Second, we show that, if A,B,C⊆ℕ have at least two elements each, then ((A+B+C)mathbinbigtriangleupℙ)(x) ≫ √x/((log x)^2). An analogous estimate is obtained for sumsets of k≥ 3 sets. The proofs use Montgomery's large sieve, Gallagher's larger sieve, a mean-value argument for a certain multiplicative function, and a submultiplicative sumset inequality.
ER -