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Statement

Printed p. 271: "Denote by $f(k)$ the smallest integer so that the product of $f(k)$ consecutive integers greater than $k$ always contain a prime greater than $k$. The well known theorem of Sylvester and Schur states $f(k)\le k$ and I proved $f(k)<\frac{3k}{\log k}$ [4]. Very much stronger results have recently been proved by Jutila, Ramachandra and Shorey [15], they showed (improving previous results of Tijdeman)

$$f(k)<\frac{c_1k\log\log\log k}{\log k\log\log k}. \tag{1}$$

(1) is certainly very far from the 'truth'. It seems sure that $f(k)=o(k^\epsilon)$ and probably $f(k)<c_1(\log k)^{c_2}$ (the $c$'s are absolute constants not necessarily the same if they have the same index). These conjectures are inaccessible at present and I have nothing to contribute towards their solution."

The reference list (printed p. 282) resolves [4] to Erdős, On consecutive integers, Nieuw Arch. voor Wisk. 3 (1955), 124--128, and [15] to three papers: M. Jutila, On numbers with large prime factors II, "will appear in the Indian J. Math."; K. Ramachandra and T. N. Shorey, On gaps between numbers with a large prime factor, Acta Arithmetica 24 (1973), 99--111; and T. N. Shorey, On gaps between numbers with a large prime factor II, Acta Arith. 25 (1974), 365--373.

Source. P. Erdős, Problems and results on consecutive integers, Publ. Math. Debrecen 23 (1976), no. 3--4, 271--282, DOI 10.5486/pmd.1976.23.3-4.15 (Crossref record read); the twelve-page scan read for this page (printed pp. 271--282 = PDF pp. 1--12, no text layer); display (1) on printed p. 271 (PDF p.

1. and the references on p. 282 (PDF p. 12), read on the page images on 2026-09-18.

Read depth. Claims checked for the passage as Erdős's report: the display and its attribution were read clause by clause on the page image. This is an attestation, not a proof: the three papers of [15] are not held here, so the bound's exact statement and hypotheses in those papers were not compared with (1). The 1955 paper states its Theorem 1 with an unspecified constant $c_1>1$; the constant $3$ is this survey's restatement.

Proof pointer

None on the page; the proofs are in the papers of [15].

Dependencies

The results of Jutila, Ramachandra and Shorey as attested; Tijdeman's earlier bounds are named without a reference.

Bears on

• Problem 961: the record upper bound for $f(k)$ and Erdős's expectation of the true order, quoted second-hand since the underlying papers are not held.