OP-00945 Open

Every integer as a sum of two very smooth numbers

Let $\varepsilon > 0$. The integers up to $N$ whose prime factors are all at most $N^{\varepsilon}$ have positive density, depending on $\varepsilon$, but the problem seems close to binary problems such as Goldbach's. Green believes the record is still Balog's: every large $n$ is a sum of two integers with all prime factors at most $n^{4/(9\sqrt{e}) + \varepsilon}$, where $4/(9\sqrt{e}) \approx 0.2695$. Wooley observed that a prime $p \equiv 3 \pmod 4$ that is a sum of two $p^{\varepsilon}$-smooth numbers has a quadratic non-residue of size at most $p^{\varepsilon}$, so the problem for every $\varepsilon$ is at least as hard as Vinogradov's least quadratic non-residue problem for such primes.

Problem. Can every $n \le N$ be written as $a + b$ where neither $a$ nor $b$ has a prime factor larger than $N^{\varepsilon}$?

Source: P. Erdős, asked several times, e.g. in Problems and results on consecutive integers, Publ. Math. Debrecen 23 (1976), and Some new problems and results in number theory, Lecture Notes in Math. 938 (1982); Erdős problem #334. Listed as Problem 59 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
Keywords: binary additive problems, Erdős problems, friable integers, smooth numbers
Literature checked: Oct. 26, 2026 by clemens-admin

References

@article{erdos1976consecutive,
  author = {Erd{\H o}s, P.},
  title = {Problems and results on consecutive integers},
  journal = {Publicationes Mathematicae Debrecen},
  volume = {23},
  number = {3-4},
  pages = {271--282},
  year = {1976},
  doi = {10.5486/PMD.1976.23.3-4.15}
}

@incollection{erdos1982mysore,
  author = {Erd{\H o}s, Paul},
  title = {Some new problems and results in number theory},
  booktitle = {Number Theory (Mysore, 1981)},
  series = {Lecture Notes in Mathematics},
  volume = {938},
  pages = {50--74},
  publisher = {Springer},
  year = {1982},
  doi = {10.1007/BFb0097173}
}

@article{balog1989,
  author = {Balog, A.},
  title = {On additive representation of integers},
  journal = {Acta Mathematica Hungarica},
  volume = {54},
  number = {3-4},
  pages = {297--301},
  year = {1989},
  doi = {10.1007/BF01952060}
}

@misc{erdosproblems334,
  author = {Bloom, Thomas F.},
  title = {Erd{\H o}s Problem \#334},
  howpublished = {\url{https://www.erdosproblems.com/334}},
  note = {Accessed 2026-10-26}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 59; version of January 2026, most recent update December 2025},
  year = {2026}
}

Citing this problem

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OP-00945, "Every integer as a sum of two very smooth numbers", clopen.solutions.