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

Half of all positive integers have an odd number of prime factors counted with multiplicity, so one expects about half of the numbers $p - 2$, with $p$ prime, to do so as well. Nobody can prove that this happens even infinitely often. The question is a weak form of the twin prime conjecture, and it runs straight into the parity problem of sieve theory.

Problem

For a positive integer $n$ let $\Omega(n)$ be the number of prime factors of $n$ counted with multiplicity, and let $\lambda(n) = (-1)^{\Omega(n)}$ be the Liouville function. Are there infinitely many primes $p$ such that $\Omega(p - 2)$ is odd, that is, $\lambda(p - 2) = -1$?

For example $p = 5$ and $p = 7$ qualify ($p - 2 = 3$ and $5$ are prime), while $p = 11$ does not ($9 = 3^2$). Green remarks that the same question with $p - 1$ in place of $p - 2$ is perhaps more natural; it is also open (for the opposite parity, see below).

Background

If $p - 2$ is prime then $\Omega(p - 2) = 1$ is odd, so infinitely many twin primes would answer the question. The integers with an odd number of prime factors have density $1/2$, so the question "only" asks for infinitely many primes in a set of density $1/2$, shifted by $2$. The difficulty is the parity phenomenon: sieve methods cannot distinguish integers with an odd number of prime factors from those with an even number. Chen's theorem (1966) gives infinitely many primes $p$ for which $p + 2$ has at most two prime factors, but cannot tell whether that number is one or two.

What is known

• Pintz (2010), using the Goldston–Pintz–Yıldırım method, proved that there is an even $d$ with $0 < |d| \le 16$ such that $\lambda(p + d) = -1$ for infinitely many primes $p$; but the method does not say which $d$, so the specific shift $d = -2$ asked about here is not covered.
• Pintz also showed that for some $d$ with $0 < |d| \le 30$ and some odd $b$ bounded by an absolute constant, infinitely many numbers $p + d$ have exactly $b$ prime factors, all larger than $p^c$.
• Pintz notes that even the Elliott–Halberstam conjecture is not known to give $\lambda(p + d) = -1$ infinitely often for a single prescribed shift $d$.
• OpenAI (September 2026) posted a proof that $\mu(p - 1) = 1$ for infinitely many primes $p$: $p - 1$ is squarefree with an even number of prime factors. This breaks the parity barrier for the shift $-1$, but for the even parity rather than the odd one; the preprint is unrefereed and not formalised.

What would count as progress

• A proof that $\lambda(p - 2) = -1$ for infinitely many primes $p$, or the same for $p - 1$.
• Narrowing Pintz's range of shifts: for example, showing that $\lambda(p + d) = -1$ infinitely often for at least one of $d = 2$ and $d = -2$.
• Adapting the September 2026 argument for $\mu(p - 1) = 1$ to an odd number of prime factors, or to the shift $-2$.
• A proof conditional on a clearly stated standard hypothesis about primes in arithmetic progressions, with the dependence made explicit.
• A formal proof of the Lean statement green_64 in GreensOpenProblems/64.lean (google-deepmind/formal-conjectures), or of its variant green_64.variants.p_sub_one.

References

1. J. Pintz, An approximation to the twin prime conjecture and the parity phenomenon, Indag. Math. 26 (2015), 883–896; arXiv:1004.1065.
2. OpenAI, Prime predecessors with an even number of prime factors, preprint (2026).

Source

Imported from Ben Green's 100 open problems 64, curated by Ben Green. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.