Erdős's conjecture on prime factors of backward shifts

This is Erdős’s 1979 conjecture on finding infinitely many shifts whose prime-factor counts are uniformly small. It has two assertions, involving distinct factors and factors counted with multiplicity; neither is fully settled.

Known results

• Lau proved that infinitely many $n$ satisfy $\omega(n-k)\leq\Omega(n-k)\leq C\log k$ for every $1<k<n$.
• A stronger $O(1)$ refinement of the $\log k/\log\log k$ bound was disproved: some backward shift has at least $\log k/\log\log k+c\log k/(\log\log k)^2$ distinct prime factors for all sufficiently large $n$.
• An earlier 2025 result gave only a linear bound $\Omega(n-k)\leq Ck$.
• The new logarithmic theorem improves that bound by a factor of roughly $k/\log k$.

April 2026 proposed obstruction

The paper conjectures that some shifts eventually have $\omega(n-k)>(1-\varepsilon)\log k$, which would disprove Erdős’s first assertion, and gives a conditional theorem in that direction. This is not an unconditional counterexample; the $\Omega$ assertion is not resolved.

Current status (as of August 2026): A logarithmic upper bound is proved, while the stated $\omega$ assertion remains neither proved nor unconditionally disproved, and the stated $\Omega$ assertion remains open.