Numbers of the form \((1^{\ell}+q^{\ell})(2^{\ell}+q^{\ell})\cdots(n^{\ell}+q^\ell)\) that are not powerful. (Nombres de la forme \((1^{\ell}+q^{\ell})(2^{\ell}+q^{\ell})\dots(n^{\ell}+q^{\ell})\) qui ne sont pas puissants.) (English. French summary) Zbl 1384.11060
Summary: Let \(q\) be a positive integer. Recently, C. Niu and W. Liu [J. Number Theory 180, 403–409 (2017; Zbl 1384.11059)] proved that, if \(n\geq \max \{q, 1198 - q \}\), then the product \((1^3 + q^3)(2^3 + q^3) \cdots(n^3 + q^3)\) is not a powerful number. In this note, we prove (1) that, for any odd prime power \(\ell\) and \(n \geq \max \{q, 11 - q \}\), the product \((1^\ell + q^\ell)(2^\ell + q^\ell) \cdots(n^\ell + q^\ell)\) is not a powerful number, and (2) that, for any positive odd integer \(\ell\), there exists an integer \(N_{q, \ell}\) such that, for any positive integer \(n \geq N_{q, \ell}\), the product \((1^\ell + q^\ell)(2^\ell + q^\ell) \cdots(n^\ell + q^\ell)\) is not a powerful number.
MSC:
| 11D25 | Cubic and quartic Diophantine equations |
Citations:
Zbl 1384.11059References:
| [1] | Amdeberhan, T.; Medina, L. A.; Moll, V. H., Arithmetical properties of a sequence arising from an arctangent sum, J. Number Theory, 128, 1807-1846 (2008) · Zbl 1149.11007 |
| [2] | Chen, Y.-G.; Gong, M.-L., On the products \((1^\ell + 1)(2^\ell + 1) \cdots(n^\ell + 1)\) II, J. Number Theory, 144, 176-187 (2014) · Zbl 1319.11003 |
| [3] | Chen, Y.-G.; Gong, M.-L.; Ren, X.-Z., On the products \((1^\ell + 1)(2^\ell + 1) \cdots(n^\ell + 1)\), J. Number Theory, 133, 2470-2474 (2013) · Zbl 1319.11004 |
| [4] | Cilleruelo, J., Squares in \((1^2 + 1)(2^2 + 1) \cdots(n^2 + 1)\), J. Number Theory, 128, 2488-2491 (2008) · Zbl 1213.11057 |
| [5] | Cilleruelo, J.; Luca, F.; Quirós, A.; Shparlinski, I. E., On squares in polynomial products, Monatshefte Math., 159, 215-223 (2010) · Zbl 1273.11124 |
| [6] | Cuellar, S.; Samper, J. A., A nice and tricky lemma (lifting the exponent), Math. Reflec., 2007, 3 (2007) |
| [7] | Fang, J.-H., Neither \(\prod_{k = 1}^n(4 k^2 + 1)\) nor \(\prod_{k = 1}^n(2 k(k - 1) + 1)\) is a perfect square, Integers, 9, 177-180 (2009) · Zbl 1160.11008 |
| [8] | Golomb, S. W., Powerful numbers, Amer. Math. Mon., 77, 848-852 (1970) · Zbl 0218.10018 |
| [9] | Gürel, E., On the occurrence of perfect squares among values of certain polynomial products, Amer. Math. Mon., 123, 597-599 (2016) · Zbl 1391.11015 |
| [10] | Gürel, E., A note on the products \(((m + 1)^2 + 1) \cdots(n^2 + 1)\) and \(((m + 1)^3 + 1) \cdots(n^3 + 1)\), Math. Commun., 21, 109-114 (2016) · Zbl 1369.11004 |
| [11] | Gürel, E.; Kisisel, A. U.O., A note on the products \((1^u + 1)(2^u + 1) \cdots(n^u + 1)\), J. Number Theory, 130, 187-191 (2010) · Zbl 1220.11115 |
| [12] | Hong, S.-F.; Liu, X., Squares in \((2^2 - 1) \cdots(n^2 - 1)\) and \(p\)-adic valuation, Asian-Eur. J. Math., 3, 329-333 (2010) · Zbl 1213.11081 |
| [13] | Niu, C.-Z.; Liu, W.-X., On the products \((1^3 + q^3)(2^3 + q^3) \cdots(n^3 + q^3)\), J. Number Theory, 180, 403-409 (2017) · Zbl 1384.11059 |
| [14] | Sándor, J.; Mitrinović, D. S.; Crstici, B., Handbook of Number Theory I (2006), Springer: Springer The Netherlands · Zbl 1151.11300 |
| [15] | Spiegelhalter, P.; Vandehey, J., Squares in polynomial product sequences |
| [16] | Yang, S.-C.; Togbé, A.; He, B., Diophantine equations with products of consecutive values of a quadratic polynomial, J. Number Theory, 131, 1840-1851 (2011) · Zbl 1242.11023 |
| [17] | Zhang, Z.-F., Powers in \(\prod_{k = 1}^n(a k^{2^l \cdot 3^m} + b)\), Funct. Approx. Comment. Math., 46, 7-13 (2012) · Zbl 1309.11026 |
| [18] | Zhang, W.-P.; Wang, T.-T., Powerful numbers in \((1^k + 1)(2^k + 1) \cdots(n^k + 1)\), J. Number Theory, 132, 2630-2635 (2012) · Zbl 1255.11001 |
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