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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 7 (p. 375), quoted: "The abc-conjecture implies that (1.1) with $d>1$, $k\geq3$ and $l\geq4$ has only finitely many solutions in $n,d,k,b,y,l$."

The paper presents it (p. 375) as a refinement of Shorey's result that for $d>1$ and $l\ge4$ the abc-conjecture bounds $k$ by an absolute constant. It notes (p. 376) that an effective variant of the abc-conjecture makes the theorem effective, and that $d>1$ is necessary, since for $d=1$ and $n=1$ equation (1.1) is solvable for every $k\ge2$. The remark after the proof (p. 387) adds that for $l\ge7$ the weak abc-conjecture with $\varepsilon=1$ and constant $1$ would also serve. The result is conditional on the abc-conjecture.

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 7 on p. 375, the remarks on pp. 375--376 and 387, the proof on p. 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 p. 386 for its structure only. A second reader checked the statement, hypotheses, label and page against the print.

Proof pointer

Section 6, p. 386. Excluding $(n,d,k)=(2,7,3)$, a theorem of Shorey and Tijdeman gives $P(\Pi)>k$, hence $P(y)>k$, and Shorey's abc-conditional result bounds $k$. For fixed $k$, the identity $(j-i)(n+(k-1)d)+(k-1-j)(n+id)=(k-1-i)(n+jd)$ with (3.1) gives the three-term equation (6.5); the abc-conjecture with $\varepsilon=1/4$ bounds $l$ and the $l$th powers in (6.5), and the remaining $S$-unit equation, with $S$ the primes up to $k$, has finitely many solutions by a result of Győry (1979). Hence $n,d,b,y$ are bounded.

Dependencies

The abc-conjecture; Shorey and Tijdeman on the greatest prime factor of an arithmetic progression; Shorey (1999); finiteness of $S$-unit equations (Győry, Comment. Math. Helv. 54 (1979)).

Bears on

• Problem 672: conditionally on the abc-conjecture, with $b=1$, only finitely many primitive positive progressions with $d>1$, of any length $k\ge3$, have a product equal to an $l$th power with $l\ge4$, counting all lengths and exponents together. Conditional and finiteness only; the exponents $l=2,3$ and the case $d=1$ are outside it.