---

Claim. For an absolute constant $c_1$,

$$f(k)<c_1\frac{k\log\log\log k}{\log k\log\log k},$$

where $f(k)$ is the function of Problem 961. The bound is stated as display (1) on printed p. 271 of Erdős's survey Problems and results on consecutive integers, Publ. Math. Debrecen 23 (1976), 271--282 (result page), which says that Jutila, Ramachandra and Shorey showed it, improving earlier results of Tijdeman. Erdős's reference [15] lists three papers together: M. Jutila, On numbers with a large prime factor. II, J. Indian Math. Soc. (N.S.) 38 (1974), 125--130; K. Ramachandra and T. N. Shorey, On gaps between numbers with a large prime factor, Acta Arith. 24 (1973), no. 1, 99--111; and T. N. Shorey, On gaps between numbers with a large prime factor. II, Acta Arith. 25 (1974), no. 4, 365--373. The site credits the bound to Jutila and to Ramachandra and Shorey. How the bound divides between the three papers is not stated here: the papers' own statements and hypotheses are not recorded, so the bound and its form rest on Erdős's attestation and the site's. This page carries the first of January 1973, the year of the earliest of the three papers.

Covers. The upper bound above; the order of $f(k)$ stays open.

Depends on. No page of this wiki.

Acceptance. Refereed, through the journal publications in the Journal of the Indian Mathematical Society and Acta Arithmetica, as Erdős's survey attests them. The site labels the problem OPEN, so its commentary gives no reviewed evidence.