On a conjecture about repdigits in k-generalized Fibonacci sequences

Mersenne k-Fibonacci numbers

“…For instance, F. Luca ([11]) and D. Marques ([13]) proved that 55 and 44 are the largest repdigits (i.e., numbers with only one distinct digit in its decimal expansion) in the sequences F (2) and F (3) , respectively. Moreover, D. Marques conjectured that there are no repdigits, with at least two digits, belonging to F (k) , for k > 3.…”

“…To prove our main result we use lower bounds for linear forms in logarithms (Baker's theory) to bound n and m polynomially in terms of k. When k is small, we use the theory of continued fractions by means of a variation of a result of Dujella and Pethő to lower such bounds to cases that allow us to treat our problem computationally. For large values of k, Bravo and Luca in [1,2] developed some ideas for dealing with Diophantine equations involving k−Fibonacci numbers. However, when k is large and m = n − 2, the estimates given in [1,2] are not enough and therefore we need to get more accurate estimates to finish the job.…”

“…For large values of k, Bravo and Luca in [1,2] developed some ideas for dealing with Diophantine equations involving k−Fibonacci numbers. However, when k is large and m = n − 2, the estimates given in [1,2] are not enough and therefore we need to get more accurate estimates to finish the job. For this aim, the formula of Cooper and Howard (Lemma 1.1) will play an important role.…”

Repdigits as Euler functions of Lucas numbers

“…We next present the following lemma from [2], which is an immediate variation of the result due to Dujella and Pethő from [7], and will be the key tool used to reduce the upper bound on the variable n when we assume that n ∈ {p, p 2 } .…”

On Pillai's Problem With Tribonacci Numbers and Powers of 2

“…For equation (2), which can be seen as a variation of Pillai's problem, they proved that the only integers c having at least two representations of the form F n − 2 m are c ∈ {0, 1, −3, 5, −11, −30, 85} (here, F 1 = F 2 = 1 are identified so representations involving F 1 or F 2 do not count as distinct). Moreover, they computed all the representations of the form (2) for all these values of c.…”