Problem 891

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Statement. Let $2=p_1<p_2<\cdots$ be the primes and $k\geq 2$. Is it true that, for all sufficiently large $n$, there must exist an integer in $[n,n+p_1\cdots p_k)$ with $>k$ many prime factors?

Formulation. Prime factors are counted without multiplicity: the question asks for an $m$ in the interval with $\omega(m)>k$. This is how the source reads it. Erdős and Selfridge count distinct prime factors, and the site's remark that even the case $k=2$ is unknown requires it. Counted with multiplicity the question would be trivial: for $n>2^k$ the interval, of length $p_1\cdots p_k\ge2^k$, contains a multiple $2^km$ of $2^k$ with $m\ge2$, which has more than $k$ prime factors counted with multiplicity.

Status. Open.

Source. erdosproblems.com/891, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #891, https://www.erdosproblems.com/891.

References.

• [Po18] Pólya, Georg, Zur arithmetischen Untersuchung der Polynome. Math. Z. 1 (1918), 143-148.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1967_problems_prime_factors_consecutive_integers
• erdos_1967_problems_prime_factors_consecutive_integers / remark_p430
• polya_1918_zur_arithmetischen_untersuchung_der_polynome
• polya_1918_zur_arithmetischen_untersuchung_der_polynome / equation_14
• polya_1918_zur_arithmetischen_untersuchung_der_polynome / satz_1