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Statement

Problem B (p. 557). Let $f(x)$ count the EHS numbers (the members of the set $\mathcal S$ of Definition 2.11) up to $x$. The paper asks whether $\lim_{x\to\infty}f(x)/x$ exists and, if so, what it is.

The authors report a list of all EHS numbers up to $2^{10}$. They print the values of $f(x)/x$ at $x=100,200,300,400,500$, "correct to two decimals," as "5.5, 5.25, 5.7, 5.45, and 4.98" [sic]; a ratio $f(x)/x$ cannot exceed $1$, so the values as printed cannot be the ratios. From them they say the limit, if it exists, would be around $0.5$; they then say that the EHS numbers occur mostly in long runs of consecutive integers, which makes them believe that the asymptotic density exists and is $1$, and that Erdős, at first hesitant, came to the same view. Section 4 (p. 558) quotes Erdős's letter of 29 July 1993: he thinks that for almost all $n$ there is a prime $p\not\equiv1\pmod n$ dividing $n!+1$, but does not see how to prove it.

Problem B is not starred, so by Section 3's preamble it was raised in discussion with Erdős.

Proof pointer

An open problem; the paper proves nothing about it beyond the computed values.

Read depth

Claims checked: Problem B and the July 1993 letter quoted in Section 4 were read clause by clause on the page images of the print. The printed values were not recomputed. Nothing here is independently reviewed.

Dependencies

• Theorem 2.12: the set $\mathcal S$ is infinite.

Source. G. E. Hardy and M. V. Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly 109 (2002), no. 6, 554--559, doi:10.2307/2695445; the edition read is named on the source card.

Bears on

• Problem 1074: the problem's first question, whether $\lvert S\cap[1,x]\rvert/x$ has a limit and what it is, is Problem B with the site's $S$ equal to the paper's $\mathcal S$. The paper poses it, conjectures the value $1$ and proves nothing about it.