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Statement

Lemma 4.3 (printed p. 5). "$n!+k$ is powerful finitely often."

The lemma is one of the parts into which the paper breaks the proof of Theorem 4.1, so $k$ is the fixed integer $k\ge0$ of that theorem, and the statement is read as: for fixed $k$, only finitely many $n$ make $n!+k$ powerful. The lemma's statement does not mention the abc conjecture; its proof invokes it, so the lemma holds as proved only under the hypothesis of Theorem 4.1. The proof takes $n\ge k$ and uses $k\mid n!$, so it treats $k\ge1$; the case $k=0$ is Lemma 4.2.

Source. D. Cushing and J. E. Pascoe, Powerful numbers and the ABC-conjecture, arXiv:1611.01192v1 (3 November 2016); Lemma 4.3 on p. 5, its proof on pp. 5--6. The edition is identified in the source digest.

Read depth. Claims checked: the statement and the shape of the proof were read on the page images of the preprint; the inequalities were not checked step by step, and nothing here is independently reviewed.

Proof pointer

Pp. 5--6. For $n\ge k$ and $n!+k=x$ with $x$ powerful, the proof divides through by $k$ and applies the abc conjecture with $\varepsilon=\frac12$ to the coprime triple $n!/k+1=x/k$. The radical of the product is at most $n\#\cdot x^{1/2}$, by $\operatorname{rad}(n!)=n\#$ (Lemma 2.5, p. 3) and the radical bound for powerful numbers (Lemma 2.6, p. 3, used in the form $\operatorname{rad}(x)\le x^{1/2}$). Because $n\#$ grows only exponentially in $n$ while $x>n!$, the triple satisfies $\operatorname{rad}(abc)^{3/2}<c$ for all large $n$, which the abc conjecture allows only finitely often. The final display writes $x=c$ where the triple has $c=x/k$.

Bears on

• Problem 936: with $k=1$ the lemma gives, assuming abc, that $n!+1$ is powerful for only finitely many $n$. This is the $n!+1$ case of the problem, conditionally; the $n!-1$ case is Exercise 4.4, left to the reader.
• Problem 398: every square is powerful, so the case $k=1$ also gives, assuming abc, only finitely many solutions of $n!+1=x^2$. The problem asks whether the known solutions are the only ones, which a finiteness statement does not answer; the paper's introduction (p. 2) recalls this finiteness as already shown under abc by Overholt (its reference [2]).