Sums of Fibonacci numbers that are perfect powers
نویسندگان
چکیده
Let us denote by Fn the n-th Fibonacci number. In this paper we show that for a fixed integer y there exists at most one exponent > 0 such Diophantine equation + Fm = ya has solution (n; m; a) in positive integers satisfying n m 0, unless 2; 3; 4; 6 or 10.
منابع مشابه
Perfect Powers That Are Sums of Consecutive Cubes
Euler noted the relation 63= 33+43+53 and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular, Stroeker determined all squares that can be written as a sum of at most 50 consecutive cubes. We generalize Stroeker’s work by determi...
متن کاملOn perfect numbers which are ratios of two Fibonacci numbers ∗
Here, we prove that there is no perfect number of the form Fmn/Fm, where Fk is the kth Fibonacci number.
متن کاملAliquot sums of Fibonacci numbers
Here, we investigate the Fibonacci numbers whose sum of aliquot divisors is also a Fibonacci number (the prime Fibonacci numbers have this property).
متن کاملOn powers that are sums of consecutive like powers
1 Background The problem of cubes that are sums of consecutive cubes goes back to Euler ([10] art. 249) who noted the remarkable relation 33 + 43 + 53 = 63. Similar problems were considered by several mathematicians during the nineteenth and early twentieth century as surveyed in Dickson’sHistory of the Theory of Numbers ([7] p. 582–588). These questions are still of interest today. For example...
متن کاملOn Fibonacci Numbers Which Are Powers: I I
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 ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Quaestiones Mathematicae
سال: 2022
ISSN: ['1727-933X', '1607-3606']
DOI: https://doi.org/10.2989/16073606.2022.2109220