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Statement

Equation (1.2), its normalization $0\le\alpha<l$ and its trivial solutions are as on Theorem 3 (p. 374).

Theorem 5 (p. 375). Let $2\le k\le5$ and $l\ge3$, as in Theorem 4, and assume $l\ne4$ if $k=2$. Let $\alpha>0$.

• (i) If $k=2$, the only non-trivial solutions of (1.2) are $(x,z,\alpha)=(-1/2,1/2,l-2)$, $(-2,1,1)$, $(1,1,1)$.
• (ii) If $k=3$ or $4$, (1.2) has no non-trivial solution.
• (iii) If $k=5$, (1.2) implies $l=5$ and $\alpha\in\{3,4\}$.

The remark after it (p. 375) says the assumption $l\ne4$ for $k=2$ is necessary: infinitely many coprime positive triples $(p,q,r)$ satisfy $2p^4-q^4=r^2$, and $x=q^4/r^2$ then gives infinitely many solutions of (1.2) with $k=2$, $\alpha=1$, $l=4$.

Source. K. Győry, L. Hajdu and N. Saradha, On the Diophantine equation $n(n+d)\cdots(n+(k-1)d)=by^l$, Canad. Math. Bull. 47 (2004), no. 3, 373--388, doi:10.4153/CMB-2004-037-1; Theorem 5 and its remark on p. 375, the proof on pp. 384--385. The edition is recorded on the source card.

Read depth. Claims checked: the statement and the remark were read clause by clause against the published print, and the proof on pp. 384--385 for its structure only. A second reader checked the statement, hypotheses, label and page against the print.

Proof pointer

Section 5, pp. 384--385. Theorem 3 gives every solution for $k=2$ with $l\ge3$ prime and leaves $k=l=3,4,5$ and $k=2$, $l=8$. Theorem 8(i) excludes $k=l=3$; Theorem 9(ii) excludes $k=l=4$; Theorem 8(iii) gives $\alpha\in\{3,4\}$ for $k=l=5$. For $k=2$, $l=8$, (1.3) leads to an equation $\pm2^{\beta_0}x_0^4+2^{-\gamma/2}v^4=\pm2^{\beta_1}x_1^4$, and Lemma 2 returns only solutions already found. Bennett, Bruin, Győry and Hajdu (Proc. London Math. Soc. (3) 92 (2006), p. 292) say the proofs of Theorems 8 and 9 for $l=3$ depend on an incorrect lemma of this paper (Lemma 6); the case $k=l=3$ here uses Theorem 8(i) with $l=3$.

Dependencies

Theorem 3, Theorem 8 parts (i) and (iii), Theorem 9(ii) and Lemma 2 of the same paper.

Bears on

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