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Statement

Equation (1.1) and its hypotheses are as on Theorem 1: positive integers $n,d,y,b$, integers $l,k\ge2$, $\gcd(n,d)=1$, $P(b)\le k$ and $b$ free of $l$th powers (p. 373).

Theorem 6 (p. 375), quoted: "For fixed $k\geq3$ and $l\geq2$ with $k+l>6$, equation (1.1) has only finitely many solutions in $n,d,b,y$."

The paper notes (p. 375) that Darmon and Granville (Bull. London Math. Soc. 27 (1995), 513--543), applying Faltings' theorem, had shown this for $b=1$, $k\ge3$ and $l\ge4$ fixed, and that Theorem 6 refines their result and extends it to $b>1$. It is best possible in the sense that for fixed $k\ge3$, $l\ge2$ with $k+l\le6$, (1.1) has infinitely many solutions in each case, citing Tijdeman; and the proof shows the result also holds for solutions of (1.1) with $n<0$. The theorem gives finiteness only; it gives no bound on the solutions.

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 6 and the remarks around it on p. 375, the proof on pp. 385--386. The edition is recorded on the source card.

Read depth. Claims checked: the statement and the remarks were read clause by clause against the published print, and the proof on pp. 385--386 for its structure only. A second reader checked the statement, hypotheses, label and page against the print.

Proof pointer

Section 6, pp. 385--386. Write each term as $n+id=a_ix_i^l$ as in (3.1) (p. 377), with $a_i$ free of $l$th powers and $P(a_i)\le k$, so the $a_i$ take finitely many values; fix them. Three-term relations such as $(n+id)+(n+(i+2)d)=2(n+(i+1)d)$ give the identities (6.1)--(6.3), whose product is an equation $F(x_i,x_{i+2})=z^l$ with $F$ a binary form having enough pairwise linearly independent linear factors: three such equations multiplied for $k\ge5$, two for $k=4$ (where $l\ge3$), one for $k=3$ (where $l\ge4$). Theorem 1 of Darmon and Granville then leaves finitely many values of $x_i$ and $x_{i+2}$, hence of every $x_i$, and so of $n,d,b,y$.

Dependencies

Theorem 1 of Darmon and Granville (1995), which rests on Faltings' theorem; the factorization (3.1) of the same paper.

Bears on

• Problem 672: with $b=1$, for each fixed length $k\ge4$ and exponent $l\ge2$ with $k+l>6$, that is every $l\ge2$ when $k\ge5$ and every $l\ge3$ when $k=4$, at most finitely many primitive positive progressions of length $k$ have a product equal to an $l$th power. This is finiteness for each fixed pair $(k,l)$, not nonexistence; it does not bound the exponent or the length.