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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

MSC:
11D41 Higher degree equations; Fermat’s equation
11B25 Arithmetic progressions
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References:
[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).
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