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Source. Theorem 7, p. 458, of Kurt Mahler, On the lattice points on curves of genus 1, Proc. London Math. Soc. (2) 39 (1935), 431--466, the edition named on the source card.

Read depth. Claims checked: the statement was read clause by clause on p. 458. The paper gives no separate proof; it says the theorem follows by specializing the form and the angle in Theorem 5. Nothing here is independently reviewed.

Statement

Theorem 7 (p. 458). There is an infinite set of positive integers $k_1,k_2,k_3,\ldots$ with

$$1\le k_1<k_2<k_3<\cdots$$

such that the number of representations of $k_\nu$ in the form

$$k_\nu=pq(p+q)$$

with positive integers $p,q$ is greater than $\sqrt[4]{\log k_\nu}$.

Proof pointer

P. 458. The paper says only that Theorems 6 and 7 follow by specializing $F$ and $G$ in Theorem 5. One reading of that step: take the form $xy(x+y)$, which has only simple linear factors, and an angle with $0<A<B$, and change signs and take $\gamma<1$ as for Theorem 6.

Dependencies

Theorem 5.

Bears on

No Erdős problem in the corpus.