On the exponential diophantine equation Unx+Un+1x=Um

Let { U n } n ≥ 0 be the Lucas sequence. For integers x , n and m , we find all solutions to U n x + U n + 1 x = U m . The equation was studied and claimed to be solved completely in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but there are some computational bugs in that publication becaus...

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Bibliographic Details
Published in:The Ramanujan journal 2024, Vol.64 (1), p.153-184
Main Authors: Batte, Herbert, Ddamulira, Mahadi, Kasozi, Juma, Luca, Florian
Format: Article
Language:English
Subjects:
Combinatorics
Diophantine equation
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Logarithms
Mathematics
Mathematics and Statistics
Number Theory
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Summary:Let { U n } n ≥ 0 be the Lucas sequence. For integers x , n and m , we find all solutions to U n x + U n + 1 x = U m . The equation was studied and claimed to be solved completely in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but there are some computational bugs in that publication because of the wrong statement of Mignotte’s bound from Mignotte (A kit on linear forms in three logarithms. http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf , 2008). In this paper, the main result remains the same as in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but we focus on correcting the computational mistakes in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021), involving the application of Theorem 2.1 from Mignotte (A kit on linear forms in three logarithms. http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf , 2008).
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-023-00818-x