Florian Luca

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math.NT2025

On the number of divisors of Mersenne numbers

Vjekoslav Kovač, Florian Luca

Denote $f (n) := \sum_{1 \leq k \leq n} τ (2^{k} - 1)$ , where $τ$ is the number of divisors function. Motivated by a question of Paul Erdős, we show that the sequence of ratios $f (2 n) / f (n)$ is…

math.NT2024

On the distance between factorials and repunits

Michael Filaseta, Florian Luca

We show that if $n \geq n_{0}$ , $b \geq 2$ are integers, $p \geq 7$ is prime and $n! - (b^{p} - 1) / (b - 1) \geq 0$ , then $n! - (b^{p} - 1) / (b - 1) \geq 0.5 lo g lo g n / lo g lo g lo g n$ . Further results are obta…

math.NT2024

On the Euler function of linearly recurrence sequences

Florian Luca, Makoko Campbell Manape

In this paper, we show that if $(U_{n})_{n \geq 1}$ is any nondegenerate linearly recurrent sequence of integers whose general term is up to sign not a polynomial in $n$ , then the ineq…