On a Diophantine equation involving powers of Fibonacci numbers
نویسندگان
چکیده
منابع مشابه
On Diophantine Quadruples of Fibonacci Numbers
We show that there are only finitely many Diophantine quadruples, that is, sets of four positive integers {a1, a2, a3, a4} such that aiaj +1 is a square for all 1 ≤ i < j ≤ 4, consisting of Fibonacci numbers.
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A Diophantine m-tuple is a set of m positive integers with the property that product of any two of its distinct elements is one less then a square. In this survey we describe some problems and results concerning Diophantine m-tuples and their connections with Fibonacci numbers.
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where Fm denotes the 77?th Fibonacci number, and o > 1. Without loss of generality , we may require that t be prime. The unique solution for t 2, namely (m, c) = (12, 12)5 was given by J. H. E. Cohn [2], and by 0. Wyler [11]. The unique solution for £ = 3, namely (m9 o) = (6, 2), was given by H. London and R. Finkelstein [5] and by J. C. Lagarias and D. P. Weisser [4]. A. Petho [6] showed that ...
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with integers m1, m2; henceforth, [θ] denotes the integral part of θ. Subsequently, the range for c in this result was extended by Gritsenko [3] and Konyagin [5]. In particular, the latter author showed that (1) has solutions in integers m1, m2 for 1 < c < 3 2 and n sufficiently large. The analogous problem with prime variables is considerably more difficult, possibly at least as difficult as t...
متن کاملOn Fibonacci Powers
Fibonacci numbers have engaged the attention of mathematicians for several centuries, and whilst many of their properties are easy to establish by very simple methods, there are several unsolved problems connected to them. In this paper we review the history of the conjecture that the only perfect powers in Fibonacci sequence are 1, 8, and 144. Afterwards we consider more stronger conjecture an...
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ژورنال
عنوان ژورنال: Proceedings of the Japan Academy, Series A, Mathematical Sciences
سال: 2020
ISSN: 0386-2194
DOI: 10.3792/pjaa.96.007