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Title: | On the Diophantine equation (x+1)(k) + (x+2)(k) + ... + (2x)(k) = y(n) |
Authors: | Bérczes, Attila Pink, István Uludağ Üniversitesi/Fen-Edebiyet Fakültesi/Matematik Bölümü. Savaş, Gamze Soydan, Gökhan FWV-5620-2022 GEK-9891-2022 57206274023 23566953200 |
Keywords: | Mathematics Power sums Powers Polynomial-exponential congruences Linear forms in two logarithms Sums |
Issue Date: | 12-Jul-2017 |
Publisher: | Elsevier |
Citation: | Berczes, A. vd. (2018). ''On the Diophantine equation (x+1)(k) + (x+2)(k) + ... + (2x)(k) = y(n)''. Journal of Number Theory, 183, 326-351. |
Abstract: | In this work, we give upper bounds for n on the title equation. Our results depend on assertions describing the precise exponents of 2 and 3 appearing in the prime factorization of T-k(x) = (x + 1)(k) + (x + 2)(k) + ... + (2x)(k). Further, on combining Baker's method with the explicit solution of polynomial exponential congruences (see e.g. [6]), we show that for 2 <= x <= 13, k >= 1,y >= 2 and n >= 3 the title equation has no solutions. |
URI: | https://doi.org/10.1016/j.jnt.2017.07.020 https://www.sciencedirect.com/science/article/pii/S0022314X17302895 http://hdl.handle.net/11452/33996 |
ISSN: | 0022-314X 1096-1658 |
Appears in Collections: | Scopus Web of Science |
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