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On the resolution of the Diophantine equation Un+Um=xq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_n + U_m = x^q$$\end{document}On the resolution of the Diophantine...P. K. Bhoi et al.
被引：0
作者：
P. K. Bhoi
[1
]
S. S. Rout
[2
]
G. K. Panda
[1
]
机构：
[1] National Institute of Technology,Department of Mathematics
[2] National Institute of Technology,Department of Mathematics
关键词：
Binary recurrence sequence;
Lucas sequences;
Diophantine equation;
Linear forms in logarithms;
Primary 11B39;
Secondary 11D61;
11J86;
D O I：
10.1007/s11139-024-01002-5
中图分类号：
学科分类号：
摘要：
Suppose that (Un)n≥0\documentclass[12pt]{minimal}
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\begin{document}$$(U_{n})_{n \ge 0}$$\end{document} is a binary recurrence sequence and has a dominant root α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} with α>1\documentclass[12pt]{minimal}
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\begin{document}$$\alpha >1$$\end{document} and the discriminant D is square-free. In this paper, we study the Diophantine equation Un+Um=xq\documentclass[12pt]{minimal}
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\begin{document}$$U_n + U_m = x^q$$\end{document} in integers n≥m≥0\documentclass[12pt]{minimal}
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\begin{document}$$n \ge m \ge 0$$\end{document}, x≥2\documentclass[12pt]{minimal}
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\begin{document}$$x \ge 2$$\end{document}, and q≥2\documentclass[12pt]{minimal}
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\begin{document}$$q \ge 2$$\end{document}. Firstly, we show that there are only finitely many of them for a fixed x using linear forms in logarithms. Secondly, we show that there are only finitely many solutions in (n, m, x, q) with q,x≥2\documentclass[12pt]{minimal}
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\begin{document}$$q, x\ge 2$$\end{document} under the assumption of the abc-conjecture. To prove this, we use several classical results like Schmidt subspace theorem, a fundamental theorem on linear equations in S-units and Siegel’s theorem concerning the finiteness of the number of solutions of a superelliptic equation.
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