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Source. Section 4, p. 139, of P. Erdős and K. Mahler, On the number of integers which can be represented by a binary form, J. London Math. Soc. 13 (1938), 134--139, reprinted in Doc. Math. (2019), 475--481, as identified on the source card. Page numbers are those of the 1938 journal print.

Statement

Let $F$ be as in the standing hypotheses of the paper (an integral binary form of degree $n\ge3$ with nonzero discriminant), and let $A(u)$ be the number of integers $k$ with $1\le k\le u$ that are represented by $|F(x,y)|$ with integers $x,y$.

• By a theorem the paper attributes to Siegel, the inequality $0<|F(x,y)|\le u$ has only $O(u^{2/n})$ solutions in integers $x,y$.
• Hence $A(u)=O(u^{2/n})$, and together with Theorem 1 this gives $\liminf_{u\to\infty}A(u)/u^{2/n}>0$ and $\limsup_{u\to\infty}A(u)/u^{2/n}<\infty$.

Proof pointer

The upper bound is cited, not proved: the paper's footnote (p. 139) says Siegel's proof had not been published and refers to K. Mahler, Acta Math. 62 (1934), 92 ff. The lower bound is Theorem 1.

Dependencies

Theorem 1 and Siegel's theorem as cited. Read depth: claims checked on p. 139 of the print; the cited theorem of Siegel was not checked.

Bears on

• Problem 325: background only. The upper bound concerns values of one binary form and says nothing about sums of three $k$th powers.