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Statement

Notation. The paper does not define a coprime arithmetic progression; the introduction (p. 2) states the result for progressions $a_n=a+nd$ with $(a,d)=1$, and that reading is used here. Definition 5.1 (p. 6): if $(a_n)$ is an arithmetic progression and $a_k$, $a_{k+1}$, $a_{k+2}$ are all powerful for some $k$, then $(a_k,a_{k+1},a_{k+2})$ is a powerful triple. The definition's index set is printed incompletely, as "$(a_n)_{n\in}$" [sic].

Theorem 5.2 (printed p. 6). "Let $(a_n)$ be a coprime arithmetic progression with common difference $d$. Under the assumption of the abc-conjecture there exists [sic] only finitely many powerful triples inside $(a_n)$."

The statement does not say that the terms are positive or that $d>0$; the proof works with an increasing progression of positive terms.

Source. D. Cushing and J. E. Pascoe, Powerful numbers and the ABC-conjecture, arXiv:1611.01192v1 (3 November 2016); Definition 5.1 and Theorem 5.2 on p. 6, the proof on pp. 6--7. The edition is identified in the source digest.

Read depth. Claims checked: the statement, Definition 5.1 and the shape of the proof were read on the page images of the preprint; the chain of inequalities was not checked step by step, and nothing here is independently reviewed.

Proof pointer

Pp. 6--7. With $N=\operatorname{rad}(d)$, take a powerful triple with $a_k>N^5$. The identity $d^2+a_ka_{k+2}=a_{k+1}^2$ is an abc triple, and the radical bound for powerful numbers (Lemma 2.6, p. 3) bounds the radical of its product by $N\,(a_ka_{k+1}a_{k+2})^{1/2}$. With $\varepsilon=\frac16$ and $a_k>N^5$ this radical, raised to the power $\frac76$, is below $a_{k+1}^{119/60}<a_{k+1}^2$, so the abc conjecture leaves only finitely many such triples. That the triples with $a_k\le N^5$ are finitely many is left implicit in the print.

Bears on

• Problem 364: the progression of positive integers ($a=d=1$) is coprime, so the theorem gives, assuming abc, only finitely many triples of consecutive positive integers that are all powerful. The problem asks whether any such triple exists, which a finiteness statement does not answer. The paper (p. 6) recalls that finiteness in the integers under abc was known, and the introduction (p. 2) credits it to its reference [3]; the theorem extends it to coprime progressions.