OP-00948 Open

Primes p for which p − 2 has an odd number of prime factors

The integers with an odd number of prime factors have density $\frac12$, so the question asks for infinitely many primes in a shift by $2$ of a set of density $\frac12$; the twin prime conjecture would answer it. Green finds the same question with $p - 1$ in place of $p - 2$ perhaps more natural.

Problem. Are there infinitely many primes $p$ such that $p - 2$ has an odd number of prime factors?

Source: B. Green, 100 open problems (version of January 2026), Problem 64.

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: NT — Number Theory
Keywords: Liouville function, parity problem, shifted primes, twin primes
Literature checked: Oct. 26, 2026 by clemens-admin

References

@article{pintz2015parity,
  author = {Pintz, J{\'a}nos},
  title = {An approximation to the twin prime conjecture and the parity phenomenon},
  journal = {Indagationes Mathematicae (N.S.)},
  volume = {26},
  number = {5},
  pages = {883--896},
  year = {2015},
  doi = {10.1016/j.indag.2015.09.008},
  eprint = {1004.1065},
  archivePrefix = {arXiv}
}

@misc{openai2026primepred,
  author = {{OpenAI}},
  title = {Prime predecessors with an even number of prime factors},
  year = {2026},
  howpublished = {\url{https://github.com/openai/math/tree/main/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026}},
  note = {AI-generated preprint, 17 September 2026, not refereed; the shift $p - 1$ with even parity}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 64; 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 It will not change if the problem is solved, reclassified or withdrawn.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-00948, "Primes p for which p − 2 has an odd number of prime factors", clopen.solutions.