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Statement

Conjecture (p. 117, unnumbered, quoted). After noting the gap between the bounds of Theorems 1 and 2 and guessing that the lower bound is nearer the truth, the authors write: "In fact, we conjecture that the upper bound in (2) can be replaced by $N^\varepsilon$ (for all $\varepsilon>0$ and $N>N_2(\varepsilon)$) and, perhaps, even by $(\log N)^c$." They add that they have not been able to prove this.

The bound in (2) is the bound $3N^{3/4}\log N$ on $|\mathcal A|$ for a set $\mathcal A\subset\{1,\ldots,N\}$ with $a+a'$ squarefree for all $a,a'\in\mathcal A$ (see Theorem 2). The constant $c$ is not specified.

Proof pointer

None: the paper proves neither statement.

Read depth

Claims checked: the sentence was read on the page image of the print (p. 117). Nothing here is independently reviewed.

Dependencies

Theorem 1 and Theorem 2 give the bounds the conjecture refers to.

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

• Problem 1109: the conjecture is the problem's two questions, $f(N)\le N^{o(1)}$ and $f(N)\le(\log N)^{O(1)}$, as the authors posed them; the paper proves neither.