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Statement

Printed p. 182: "The following question can be considered: Let $a_1<a_2<\cdots<a_z\le n$ be a sequence of integers so that the products

$$\prod_{i=1}^{z}a_i^{\epsilon_i},\qquad \epsilon_i=0\ \text{or}\ 1 \tag{3}$$

are all distinct. What is the maximum of $z$? I proved that $z<\pi(n)+2n^{2/3}$ and it seems likely that $z<\pi(n)+cn^{1/2}/\log n$."

The page continues with display (4), the equal-product-length condition of Problem 786 ("a completely different question"), which the source card treats.

Source. P. Erdős, Extremal problems in number theory, Proc. Sympos. Pure Math. VIII (1965), 181--189; printed p. 182 (PDF p. 2 of the 11-page scan read for this page), read on the page image; a site key for Problem 795.

Read depth. Claims checked: the passage was read clause by clause on the page image. The bound $z<\pi(n)+2n^{2/3}$ is asserted as proved without proof on this page; by footnote 1 (printed p. 181) a result stated without reference refers to Erdős's Hungarian paper (Mat. Lapok 13 (1962), 228--255; erdos_1962_szamelmeleti_megjegyzesek_iv), whose display (5) on p. 235 states it with a proof sketch; the $cn^{1/2}/\log n$ bound is a guess.

Proof pointer

None on the page; the Hungarian paper that footnote 1 names sketches the proof of the $2n^{2/3}$ bound on its p. 235. Problem 795's page records Erdős's later papers, which prove the guessed bound and discuss the second-order term.

Dependencies

None stated.

Bears on

• Problem 795: Erdős's 1965 statement of the problem's question, with the bound he had proved and the bound he expected.
• Problem 786: display (4) on the same page is that problem's question; covered on the source card, not here.