Erdős proved that if the pairwise sums $a+b$ are all distinct aside from the trivial coincidences then\[\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0.\]This is discussed in problem C11 of Guy's collection [Gu04], in which Guy says Erdős offered \$500 for the general problem of whether, for all $h\geq 2$,\[\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/h}}=0\]whenever the sum of $h$ terms in $A$ are distinct. This was proved for $h=4$ by Nash [Na89] and for all even $h$ by Chen [Ch96b].

View the LaTeX source

This page was last edited 30 September 2025.