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Prime Predecessors with an Even Number of Prime Factors
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Abstract
We prove that there are infinitely many primes p for which $p-1$ is squarefree and has an even number of prime factors. Equivalently, there are infinitely many primes p with $\mu(p-1)=1$.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Jizhou Guo, Shifted primes with a prescribed parity of the number of prime factors (preprint, 2026-10-09). Uses this paper’s prime-slot correlations and the weighted-dilation paper’s sieve estimates to claim a lower bound of $cX/(\log X)^{300}$ for squarefree $p+h$ with either Möbius sign, uniformly for $0<|h|\le(\log X)^5$. [AI-generated per source] [family-level]
Possibly relevant literature (found by the agent, not checked)
- Janos Pintz, An approximation to the twin prime conjecture and the parity phenomenon (arXiv:1004.1065, 2010). Proves that for some even $d\le16$, $p+d$ has an odd number of prime factors for infinitely many primes $p$, a parity-type result for shifted primes related to the parity obstruction discussed in this paper's introduction.
Digests and expositions
- Perplexity / Orply, Infinitely Many Primes Have Squarefree Predecessors With Even Prime-Factor Parity (Orply, 2026-10-08). Explains the manuscript’s Möbius-sign formulation and sieve removal of balanced semiprimes, distinguishing its infinitude claim from a density assertion. [AI-generated per source]
Elsewhere
- Emergent Mind: family 011
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026,
author = {{OpenAI}},
title = {{Prime Predecessors with an Even Number of Prime Factors}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf}{OAI:Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026}},
year = {2026}
}Family 011: Prime-factor statistics of $p-1$
Proves that the normalized ordered logarithms of the prime factors of $p-1$, counted with multiplicity, converge jointly to the Poisson–Dirichlet law $\mathrm{PD}(1)$ as p ranges uniformly over primes up to x and $x\to\infty$. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than $n^{1-\varepsilon}$ totient preimages, for every ε > 0.
Other papers in this family:
- Weighted dilation graphs, smooth shifted primes and totient fibers · September 24, 2026
- The Poisson-Dirichlet law for prime predecessors · September 24, 2026