Let omega(n) denote the number of distinct prime factors and let Omega(n) count prime factors with multiplicity. Erdős's conjecture. For varepsilon>0, there are infinitely many n such that omega(n-k)≤(1+varepsilon)fraclog kloglog k for all integers 1≪_varepsilon k<n. Moreover, there are infinitely many n such that Omega(n-k)≤(1+varepsilon)fraclog klog 2 for all integers 1≪_varepsilon k<n. The first assertion is disproved by the source's later conjecture; the status of the second assertion is not resolved there.

Erdős's conjecture on prime factors of backward shifts

Let $\omega(n)$ denote the number of distinct prime factors and let $\Omega(n)$ count prime factors with multiplicity. Erdős's conjecture. For $\varepsilon>0$ , there are infinitely many $n$ such that

\omega(n-k)\leq(1+\varepsilon)\frac{\log k}{\log\log k}

for all integers $1\ll_\varepsilon k<n$ . Moreover, there are infinitely many $n$ such that

\Omega(n-k)\leq(1+\varepsilon)\frac{\log k}{\log 2}

for all integers $1\ll_\varepsilon k<n$ . The first assertion is disproved by the source's later conjecture; the status of the second assertion is not resolved there.

Sources & referencesView supporting material

Primary source

Cheuk Fung Lau, “On the Number of Prime Factors of Consecutive Integers”, arXiv:2604.15042 (2026).

analytic-number-theory prime-factors

1 week ago

Progress summary

Refreshed 4 days ago

Partially solved

A 2026 paper substantially improves the known bound, but the original conjecture remains unresolved and its proposed disproof is only conditional.

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.

Sources

No solutions have been posted yet.