Jhon J. Bravo

On a conjecture about repdigits in k-generalized Fibonacci sequences

For an integer k ≥ 2, we consider the k−generalized Fibonacci sequence (F (k) n )n which starts with 0, . . . , 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms. F. Luca [2] in 2000 and recently D. Marques [3] proved that 55 and 44 are the largest numbers with only one distinct digit (so called repdigits) in the sequences (F (2) n )n and (F (3) n )n, respectively. Further, Marques conjectured that there are no repdigits having at least 2 digits in a k−generalized Fibonacci sequence for any k > 3. In this talk, we report about some arithmetic properties of (F (k) n )n and confirm the conjecture raised by Marques. This is a joint work with Florian Luca.

ON THE LARGEST PRIME FACTOR OF THE k-FIBONACCI NUMBERS

Let P(m) denote the largest prime factor of an integer m ≥ 2, and put P(0) = P(1) = 1. For an integer k ≥ 2, let

On The diophantine equation Fn+Fm=2a

(F_{n}^{(k)})_{n\geq 2-k}

Powers in products of terms of Pell's and Pell–Lucas Sequences

be the k-generalized Fibonacci sequence which starts with 0, …, 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms. Here, we show that if n ≥ k+2, then

Repdigits as Euler functions of Lucas numbers

P(F_n^{(k)}) > 0.01\sqrt{\log n \log\log n}

FACTORIALS AND THE RAMANUJAN FUNCTION

. Furthermore, we determine all the k-Fibonacci numbers

Powers of Two in Generalized Fibonacci Sequences

F_n^{(k)}

Coincidences in generalized Fibonacci sequences

whose largest prime factor is less than or equal to 7.

Repdigits as sums of two \(k\)-Fibonacci numbers

Abstract In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, a), where Fk is the kth term of the Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms (Bakers theory) and a version of the Baker-Davenport reduction method in diophantine approximation.

Powers of two as sums of two k-Fibonacci numbers

In this paper, we consider the usual Pell and Pell–Lucas sequences. The Pell sequence is given by the recurrence un = 2un-1 + un-2 with initial condition u0 = 0, u1 = 1 and its associated Pell–Lucas sequence is given by the recurrence vn = 2vn-1 + vn-2 with initial condition v0 = 2, v1 = 2. Let n, d, k, y, m be positive integers with m ≥ 2, y ≥ 2 and gcd(n, d) = 1. We prove that the only solutions of the Diophantine equation unun+d⋯un+(k-1)d = ym are given by u7 = 132 and u1u7 = 132 and the equation vnvn+d⋯vn+(k-1)d = ym has no solution. In fact, we prove a more general result.