---

Statement

Setting (p. 258). $F(x)$ is the greatest integer $k$ for which there is a run of $k$ consecutive integers $n+1,n+2,\ldots,n+k$ with $n+k\le x$ and $d(n+1),d(n+2),\ldots,d(n+k)$ all distinct; $c_2,c_3,c_4$ are absolute positive constants.

Theorem V (p. 258). For all sufficiently large values of $x$,

$$F(x)>c_2\frac{(\log x)^{1/2}}{\log\log x}.$$

Upper bound and conjecture (p. 258). The paper says it can prove no upper bound better than

$$F(x)<\exp\Bigl\{c_3\frac{(\log x)^{1/2}}{\log\log x}\Bigr\},$$

which follows trivially from Theorem II, and conjectures that the true order of magnitude of $F(x)$ is $(\log x)^{c_4}$.

Source. P. Erdős and L. Mirsky, The distribution of values of the divisor function $d(n)$, Proc. London Math. Soc. (3) 2 (1952), 257--271; Theorem V, the upper bound and the conjecture on p. 258, the proof of Theorem V in §11, pp. 269--270. The copy read is identified on the source card.

Read depth. Claims checked: the statement, the upper bound and the conjecture were read clause by clause on the page images and the proof was read; its estimates were not re-derived. Nothing here is independently reviewed.

Proof pointer

§11, pp. 269--270, a Chinese-remainder construction. Take $k=\bigl[(\log x)^{1/2}/(2\log\log x)\bigr]$ (11.1), the first $k$ primes $p_1,\ldots,p_k$, and the first $k$ primes $q_1,\ldots,q_k$ exceeding $(\log x)^{1/2}$, with $q=q_1$. Choose $t$ so that $p_\nu^{q_\nu-1}$ divides $t+\nu$ exactly, for $1\le\nu\le k$, with $t+k\le x$; the modulus $M=p_1^{q_1}\cdots p_k^{q_k}$ is below $x^{1/2}$ (11.2). A sieve over the further conditions that no $r^{q-1}$ with $r\ne p_\nu$ prime divides $t+\nu$ shows a suitable $t$ survives. Then $q_\nu$ divides $d(t+\nu)$ while $q_\mu$ does not for $\mu\ne\nu$, so the $k$ divisor counts $d(t+1),\ldots,d(t+k)$ are distinct.

Dependencies

The Chinese remainder theorem and an elementary count; the upper bound uses Theorem II through $F(x)\le D(x)$.

Bears on

• Problem 945: the theorem is the lower bound $F(x)>c_2(\log x)^{1/2}/\log\log x$ for the problem's $F(x)$, and p. 258 carries the upper bound and the conjecture that $F(x)$ has order $(\log x)^{c_4}$. The problem asks whether $F(x)\le(\log x)^{O(1)}$; neither bound decides that question.