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Statement

Notation (printed p. 2): a number $x$ is powerful if $p\mid x$ implies $p^2\mid x$; $\operatorname{rad}(x)$ is the product of the distinct primes dividing $x$ (Definition 2.1, p. 2). The abc conjecture is stated twice: as Conjecture 1.1 (pp. 1--2), for every $\varepsilon>0$ only finitely many triples $a,b,c$ with $(a,b)=(b,c)=(a,c)=1$, $a+b=c$ and $\operatorname{rad}(abc)^{1+\varepsilon}<c$; and as Conjecture 3.1 (p. 4), with the coprimality condition reduced to $(a,b)=1$.

Theorem 4.1 (printed p. 5). "Let $k\ge0$. Assuming the abc conjecture, there are finitely many $x$ such that $x$ is a powerful number and $|x-n!|\le k$"

The print does not quantify $n$; the introduction (p. 2) reads the result as "for a fixed $k$, assuming the abc conjecture, $n!+k$ is a powerful number only finitely often", so the statement is read here with $n$ ranging over all positive integers: for each fixed $k\ge0$ there are only finitely many pairs $(n,x)$ with $x$ powerful and $|x-n!|\le k$. Since each $n$ admits at most $2k+1$ such $x$, this is the same as saying that only finitely many factorials have a powerful number within distance $k$.

Source. D. Cushing and J. E. Pascoe, Powerful numbers and the ABC-conjecture, arXiv:1611.01192v1 (3 November 2016); Theorem 4.1 on p. 5, with Lemmas 4.2 and 4.3 on p. 5 and Exercise 4.4 on p. 6. The edition is identified in the source digest.

Read depth. Claims checked: the statements of Theorem 4.1, Lemma 2.6, Lemma 4.2, Lemma 4.3 and Exercise 4.4 were read clause by clause on the page images of the preprint; the proofs were read for structure only, and nothing here is independently reviewed.

Proof pointer

The paper says it breaks the proof "into three lemmas" (p. 5); what follows is two lemmas and an exercise.

• Lemma 4.2 (p. 5): $n!$ is powerful only finitely often. This case needs no conjecture; the proof uses Bertrand's postulate and states that $n!$ is not powerful for $n\ge7$, the smaller cases being checked by hand. Its divisibility claims are garbled as printed (they assert both $p\mid(2n)!$ and $p\nmid(2n)!$).
• Lemma 4.3 (pp. 5--6): $n!+k$ is powerful only finitely often, by the abc conjecture with $\varepsilon=\frac12$.
• Exercise 4.4 (p. 6): $n!-k$ is powerful only finitely often. It is left to the reader, so the half of the theorem for powerful numbers below $n!$ has no proof in the paper.

Dependencies

Lemma 4.3 bounds the radical of a powerful number by Lemma 2.6 (p. 3), printed as $\operatorname{rad}(x)<x^{1/2}$ for powerful $x$; its proof (p. 4) gives $\operatorname{rad}(x)^2\mid x$, hence only $\operatorname{rad}(x)\le x^{1/2}$, with equality exactly when $x$ is the square of a squarefree number (for instance $x=1$ or $x=4$). The proof of Lemma 4.3 uses only the non-strict form. It also uses $\operatorname{rad}(n!)=n\#$, the product of the primes up to $n$ (Lemma 2.5 with Definition 2.4, p. 3).

Bears on

• Problem 936: with $k=1$ the theorem gives, assuming abc, that $n!+1$ and $n!-1$ are powerful for only finitely many $n$, the factorial half of the problem conditionally; the $n!+1$ case is proved as Lemma 4.3, while the $n!-1$ case rests on Exercise 4.4, which the paper leaves to the reader. The theorem says nothing about $2^n\pm1$.