A composite number in a sumset below the square-root threshold
Ostmann's inverse Goldbach problem asks whether there are infinite sets $A, B$ of positive integers such that $A + B$ coincides with the set of primes from some point on. By work of Elsholtz, a positive answer to the question below would show that there are no such sets. Green and Harper related the question to the inverse large sieve problem.
Problem. Let $A, B \subset \{1, \dots, N\}$ with $|A| = |B| = N^{0.49}$. Must the sumset $A + B$ contain a composite number?
Source: B. Green and A. J. Harper, Inverse questions for the large sieve, Geom. Funct. Anal. 24 (2014), Conjecture 1.9 (stated there for every exponent δ > 0); motivated by Ostmann's inverse Goldbach problem. Listed as Problem 58 in B. Green, 100 open problems (version of January 2026).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- NT — Number Theory
- Secondary
- CO
- Keywords
- inverse Goldbach problem, large sieve, primes, sumsets
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{green2014harper,
author = {Green, Ben and Harper, Adam J.},
title = {Inverse questions for the large sieve},
journal = {Geometric and Functional Analysis},
volume = {24},
number = {4},
pages = {1167--1203},
year = {2014},
doi = {10.1007/s00039-014-0288-1},
eprint = {1311.6176},
archivePrefix = {arXiv}
}
@book{ostmann1956,
author = {Ostmann, Hans-Heinrich},
title = {Additive Zahlentheorie. Erster Teil},
series = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
volume = {7},
publisher = {Springer},
year = {1956},
doi = {10.1007/978-3-662-00842-3}
}
@article{elsholtz2001,
author = {Elsholtz, Christian},
title = {The inverse {G}oldbach problem},
journal = {Mathematika},
volume = {48},
number = {1-2},
pages = {151--158},
year = {2001},
doi = {10.1112/S0025579300014406}
}
@article{elsholtz2015harper,
author = {Elsholtz, Christian and Harper, Adam J.},
title = {Additive decompositions of sets with restricted prime factors},
journal = {Transactions of the American Mathematical Society},
volume = {367},
number = {10},
pages = {7403--7427},
year = {2015},
doi = {10.1090/S0002-9947-2014-06384-8}
}
@misc{openai2026ostmann,
author = {{OpenAI}},
title = {The additive indecomposability of the primes},
year = {2026},
howpublished = {\url{https://github.com/openai/math/tree/main/preprints/the-additive-indecomposability-of-the-primes-September-24-2026}},
note = {AI-generated preprint, 24 September 2026, not refereed; proves Ostmann's statement, not the question here}
}
@misc{croot2026sharp,
author = {Croot, Ernie and Mao, Junzhe and Pohoata, Cosmin and Yip, Chi Hoi},
title = {A sharp inverse theorem for the quadratic large sieve},
year = {2026},
eprint = {2607.15311},
archivePrefix = {arXiv}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 58; version of January 2026, most recent update December 2025},
year = {2026}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00944, "A composite number in a sumset below the square-root threshold", clopen.solutions.