Polynomial values of products of terms from an arithmetic progression
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Polynomial values of products of terms from an arithmetic progression L. Hajdu1 · Á. Papp1 Received: 3 March 2020 / Accepted: 19 April 2020 © The Author(s) 2020
Abstract Products of terms of arithmetic progressions yielding a perfect power have been long investigated by many mathematicians. In the particular case of consecutive integers, various finiteness results are known for the polynomial values of such products. In the present paper we consider generalizations of these result in various directions. Keywords Arithmetic progressions · Products of terms · Polynomial values Mathematics Subject Classification 11D41
1 Introduction A classical result of Erd˝os and Selfridge [11] says that the product of consecutive positive integers is never a perfect power, that is, the equation x(x + 1) · · · (x + k − 1) = y n
(1)
has no solutions in positive integers x, k, y, n with k ≥ 2 and n ≥ 2. This result and also Eq. (1) has been generalized into various directions. Here we only mention those directions and results which are important from our viewpoint.
Communicated by Adrian Constantin. L. Hajdu was supported in part by the NKFIH Grants 115479, 128088, and 130909, and the Projects EFOP-3.6.1-16-2016-00022 and EFOP-3.6.2-16-2017-00015 co-financed by the European Union and the European Social Fund. Á. Papp was supported by the ÚNKP-19-3 New National Excellence Program of the Ministry for Innovation and Technology.
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L. Hajdu [email protected] Á. Papp [email protected]
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Institute of Mathematics, University of Debrecen, P.O. Box 400, Debrecen 4002, Hungary
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L. Hajdu, Á. Papp
The first extension of the problem we mention is when on the left hand side of (1), we omit a term from the product, that is, we consider the equation x(x + 1) · · · (x + j − 1)(x + j + 1) · · · (x + k − 1) = y n in positive integers x, k, y, n with k ≥ 2 and n ≥ 2, where 0 ≤ j ≤ k − 1. Confirming a conjecture of Erd˝os and Selfridge, Saradha and Shorey [20,21] proved that the only solutions of the above equation are given by 4! = 23 , 3
6! = 122 , 5
10! = 7202 . 7
The second direction of extensions we mention (which probably attracted the most attention) is when instead of products of consecutive integers one takes products of terms of an arithmetic progression. More precisely, one considers the equation x(x + d) · · · (x + (k − 1)d) = y n in positive integers x, d, k, y, n with k ≥ 2, n ≥ 2 with gcd(x, d) = 1. Under certain (mild, necessary) conditions Darmon and Granville [9] proved that for fixed k and n, this equation has only finitely many solutions in x, d, y. (See also Gy˝ory, Hajdu and Saradha [14] for a further generalization.) Recently, Bennett and Siksek [2] proved that if k is large enough, then this equation has only finitely many solutions in x, d, y, n. On the other hand, for small values of k, namely for k < 35, a result of Gy˝ory, Hajdu and Pintér [13] in accordance with a conjecture of Erd˝os says that (under certain trivial necessary restrictions) this equation has no solutions at all. We also men
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