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Erdos's Conjecture on Consecutive Integers Free of Certain Prime Factors

Let $n_k$ be the least $n > 2k$ such that $(n-k)(n-k+1)\cdots(n-1)$ has no prime factor in $(k, 2k)$. Erdos conjectured a superpolynomial lower bound; for all large $k$, $n_k > e^{\log^2 k / (20 \log\log k)}$.

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number-theoryJun 18, 2026Significance 15/100Registry: unreviewed

Erdos's Conjecture on Consecutive Integers Free of Certain Prime Factors

Prior state unknown→proved

Let $n_k$ be the least $n > 2k$ such that $(n-k)(n-k+1)\cdots(n-1)$ has no prime factor in $(k, 2k)$. Erdos conjectured a superpolynomial lower bound; for all large $k$, $n_k > e^{\log^2 k / (20 \log\log k)}$.

Source authenticatedErdos's Conjecture on Consecutive Integers Free of Certain Prime Factors →
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Let $n_k$ be the least $n > 2k$ such that $(n-k)(n-k+1)\cdots(n-1)$ has no prime factor in $(k, 2k)$. Erdos conjectured a superpolynomial lower bound; for all large $k$, $n_k > e^{\log^2 k / (20 \log\log k)}$.

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