Erdős and Sárközy on prime factors of consecutive integers

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theorem_2: For all sufficiently large k, the least n with P(n+i) greater than k for every i from one through k exceeds k to the power 5/2 divided by sixteen.

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Paul Erdős and András Sárközy, On the prime factors of $\binom{n}{k}$ and of consecutive integers, Utilitas Mathematica 16 (1979), 197–215. The edition read is the nineteen-page scan in the Rényi Institute's Erdős archive. No license or usage notice was found in the scan on 2026-10-15, and no separate rights record was established. No file of this source is held, since no license on record permits redistribution.

For Problem 962, the relevant result is Theorem 2, p. 198. The paper defines $n_k$ as the least starting integer $n$ for which every integer $n+1,\ldots,n+k$ has largest prime factor greater than $k$. The theorem gives $n_k>k^{5/2}/16$ for all sufficiently large $k$. This is a uniform statement about the first such block, not a statement about almost every interval. It is a published lower bound relevant to the problem's smooth-number formulation; it does not establish the conjectured subpower scale.

Read status: claims checked. The definition on p. 197 and Theorem 2 on p. 198 were checked against the printed page images. The proof is in Section 3, beginning on p. 212; it has not been verified here. The separate binomial-coefficient result, Theorem 1, is not consumed by this card.

Bears on. #962.