Short-block conclusion without a large prime factor

Establish whether the conclusion of Lemma 5.1 holds for all but o(X) integers n≤X without assuming that the largest prime factor P(n) exceeds n^{1-α}.

Background

Lemma 5.1 shows that, when P(n)>n^{1-α}, every two-block factorial-square representation satisfying the paper’s hypotheses has both block lengths smaller than n^ε. The proof relies on writing n=ap with a small denominator and exploiting the resulting rational relation between the two blocks.

The authors leave unresolved whether an analogous short-block conclusion holds for almost all integers n≤X when no large-prime-factor hypothesis is imposed.