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Statement

Remark (pp. 117--118, unnumbered). Let $1\le a_1<a_2<\cdots<a_k\le N$ and $1<b_1<b_2<\cdots<b_l\le N$ (the second chain begins with a strict inequality in the print) be two sequences of integers such that all the sums $a_i+b_j$, $1\le i\le k$, $1\le j\le l$, are squarefree. The paper states:

• their method gives $kl<N^{3/2+\varepsilon}$;
• they can show that $kl/N\to\infty$ is possible, and have no satisfactory upper bound for $kl$;
• there is an absolute constant $c$ such that $k>cN$, $l\to\infty$ is possible, and perhaps $l$ must then be less than $\log N$ or $(\log N)^c$.

The quantifier on $\varepsilon$ and the range of $N$ in the first statement are not printed.

On p. 117 the paper also says that similar but slightly more complicated methods give analogous results for $k$-th power free numbers; it states none of them.

Proof pointer

None: the paper gives no proof or construction for these statements.

Read depth

Claims checked: the paragraph was read on the page images of the print (pp. 117--118). Nothing here is independently reviewed.

Dependencies

None in the corpus.

Source. P. Erdős and A. Sárközy, On divisibility properties of integers of the form $a+a'$, Acta Math. Hungar. 50 (1987), no. 1--2, 117--122, doi:10.1007/BF01903370; the edition read is named on the source card.

Bears on

No Erdős problem in this wiki asks the two-sequence question. The page of Problem 1109 notes that G. N. Sárközy later extended that problem to sums $A+B$; this remark concerns sums of that kind and decides nothing about Problem 1109.