zbMATH — the first resource for mathematics

On the product of consecutive elements of an arithmetic progression. (English) Zbl 0582.10011
P. Erdős and J. L. Selfridge [Ill. J. Math. 19, 292-301 (1975; Zbl 0295.10017)] proved that the product of \(k\geq 2\) consecutive integers cannot be a proper power and asked for an analogous result concerning the product of consecutive integers of an arithmetic progression. Using the method of the above-mentioned paper, the author proves that the product \((n+d)(n+2d)\cdot \cdot \cdot (n+kd)\) cannot be a proper power provided k is sufficiently large; in fact, for all cases, the largest k is \[ \leq \quad \max \{3\cdot 10^ 4, (3/2) \exp [d(d+2)(d+1)^{1/3}]\}. \]
Reviewer: E.L.Cohen

11D41 Higher degree equations; Fermat’s equation
11B25 Arithmetic progressions
Full Text: DOI EuDML
[1] Sylvester, J. J.: On arithmetic series. Messenger Math.21, 1-19 and 87-120 (1982).
[2] Rosser, J. B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Illinois J. Math.,6, 64-94 (1962). · Zbl 0122.05001
[3] Erd?s, P., Selfridge, J. L.: The product of consecutive integers is never a power. Illinois J. Math.,19, 292-301 (1975). · Zbl 0295.10017
[4] Langevin, M.: Plus grand facteur premier d’entiers en progression arithm?tique, S?minaire Delange-Pisot-Poitou. 18e ann?e. Fasc. 1, Exp. 3 (1977).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.