Treffer: On a Problem of Sidon in Additive Number Theory, and on some Related Problems: On a problem of Sidon in additive number theory, and on some related problems

For a given positive integer \(n\) the authors consider sets of distinct positive integers \(a_1,a_2,...,a_r\) not exceeding \(n\) such that the sums \(a_i+a_j (1 \leq i \leq j \leq r)\) are all different. Let \(\Phi(n)\) denote the maximum value which \(r\) can have for any such set. Then \(\Phi(n) < n^{1/2}+2n^{1/4}\) for any \(n\). On the other hand, if \(n = p^{2k}+p^k+1\), where \(k\) is a positive integer and \(p\) is a prime, then \(\Phi(n) > n^{1/2}\) [cf. \textit{J. Singer}, Trans. Am. Math. Soc. 43, 377--385 (1938; Zbl 0019.00502)]. For the addendum see Zbl 0061.07302