OP-01040 Needs verification

Positive density of Hofstadter's ab − 1 set

Needs verification: Answered after Green's list for positive lower density (Korsky, 2026); held until the record of the solution is listed here.

Note: Positive lower density is proved (Korsky, 2026); whether the natural density exists is open.

Submit a result

Let $A$ be the smallest set of positive integers that contains $2$ and $3$ and contains $a_1 a_2 - 1$ whenever $a_1, a_2 \in A$. Its elements form OEIS sequence A005244. Green expects the answer to be yes, and notes that a proof may need computation, since the analogous statement fails when $2$ and $3$ are replaced by, say, $9$ and $10$.

Problem. Does $A$ have positive density?

Source: D. Hofstadter, as reported by P. Erdős, Problems and results on combinatorial number theory III (1977); Erdős problem #424. Listed as Problem 63 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: density, Erdős problems, integer sequences, multiplicative recurrences

References

@incollection{erdos1977problems,
  author = {Erd{\H o}s, Paul},
  title = {Problems and results on combinatorial number theory, {III}},
  booktitle = {Number Theory Day (Proc. Conf., Rockefeller Univ., New York, 1976)},
  series = {Lecture Notes in Mathematics},
  volume = {626},
  pages = {43--72},
  publisher = {Springer},
  year = {1977},
  doi = {10.1007/BFb0063064}
}

@misc{erdosproblems424,
  author = {Bloom, Thomas F.},
  title = {Erd{\H o}s Problem \#424},
  howpublished = {\url{https://www.erdosproblems.com/424}},
  note = {Accessed 2026-10-27}
}

@misc{korsky2026hofstadter,
  author = {Korsky, Samuel},
  title = {Positive lower density for {H}ofstadter's $ab-1$ problem},
  year = {2026},
  eprint = {2608.07910},
  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 63; 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 It will not change if the problem is solved, reclassified or withdrawn.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-01040, "Positive density of Hofstadter's ab − 1 set", clopen.solutions.