A variant of the Nagell-Ljunggren superelliptic equation. (English) Zbl 07833716
Summary: In this note we study the finiteness of solutions of the exponential Diophantine equation
\[
\left(\frac{x^d-1}{x-1}\right)^2-\frac{x^d(x^{d-1}-1)}{x-1}=y^m
\]
in rational integers \(x\), \(y\), \(d\ge 1\) and \(m\ge 2\).
MSC:
| 11D61 | Exponential Diophantine equations |
References:
| [1] | Bagchi, Bhaskar, Parametric restrictions on quasi-symmetric designs, European Journal of Combinatorics, 99, 103434, 1-22 (2022) · Zbl 1476.05019 |
| [2] | Baker, A., Bounds for the solutions of hyperelliptic equation, Proc. Camb. Phil. Soc., 65, 439-444 (1969) · Zbl 0174.33803 · doi:10.1017/S0305004100044418 |
| [3] | A. Baker, Transcendental Number Theory, Cambridge University Press (2nd edn 1979). · Zbl 0297.10013 |
| [4] | M. A. Bennett and A. Levin, The Nagell-Ljunggren equation via Runge’s method, Monatsh Math 177 (2015), 15-31. doi:10.1007/s00605-015-0748-1. · Zbl 1395.11060 |
| [5] | Bridy, A.; Oliver, RJL; Shallit, A.; Shallit, J., The generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, Experimental Mathematics, 28, 4, 428-439 (2019) · Zbl 1475.11050 · doi:10.1080/10586458.2017.1419391 |
| [6] | Y. Bugeaud and M. Mignotte, L’equation de Nagell-Ljunggren \(\frac{x^n-1}{x-1}=y^q,\) Enseign.Math. 48 (2002), 147-168. · Zbl 1040.11015 |
| [7] | W. Ljunggren, Noen setninger om ubestemte likninger av formen \(\frac{x^n-1}{x-1}=y^q,\) Norsk. Mat. Tidsskrift, 25 (1943), 17-20 ; Collected Papers of W. Ljunggren edited by P. Riebenboim, Volume 1, \(\# 14\), p. 363-366. |
| [8] | Ljunggren, W., On the irreducibility of certain trinomials and quadrinomials, Math. Scand., 8, 65-70 (1960) · Zbl 0095.01305 · doi:10.7146/math.scand.a-10593 |
| [9] | Schinzel, A.; Tijdeman, R., On the equation \(y^m=P(x)\), Acta Arith., 31, 199-204 (1976) · Zbl 0303.10016 · doi:10.4064/aa-31-2-199-204 |
| [10] | T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press (1986). · Zbl 0606.10011 |
| [11] | T. N. Shorey, Exponential diophantine equations involving products of consecutive integers and related equations, Number Theory ed. by R. P. Bambah, V. C. Dumir and R. J. HansGill, Hindustan Book Agency (1999), 463-495. · Zbl 0958.11026 |
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