1. On sums of two Fibonacci numbers that are powers of numbers with limited hamming weight.
- Author
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Vukusic, Ingrid and Ziegler, Volker
- Subjects
HAMMING weight ,DIOPHANTINE equations ,LOGICAL prediction ,HAMMING distance - Abstract
In 2018, Luca and Patel conjectured that the largest perfect power representable as the sum of two Fibonacci numbers is 3864
2 = F36 + F12 . In other words, they conjectured that the equation has no solutions with a ≥ 2 and ya > 38642 While this is still an open problem, there exist several partial results. For example, recently Kebli, Kihel, Larone and Luca proved an explicit upper bound for ya , which depends on the size of y. In this paper, we find an explicit upper bound for ya , which only depends on the Hamming weight of y with respect to the Zeckendorf representation. More specifically we prove the following: If y = Fn 1 + · · · + Fnk and equation (*) is satisfied by y and some non-negative integers n, m and a ≥ 2, then Here, ϵ > 0 can be chosen arbitrarily and C(ϵ) is an effectively computable constant. [ABSTRACT FROM AUTHOR]- Published
- 2024
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