For every positive integer $n$ let $p\left(n\right)$ be the largest prime number $p\le n$. Given a positive integer $n=n_1$, we study the positive integer $r=R\left(n\right)$ such that if we define recursively $n_{i+1}=n_i-p\left(n_i\right)$ for $i\ge 1$, then $n_r$ is a prime or $1$. We obtain upper bounds for $R\left(n\right)$ as well as an estimate for the set of $n$ whose $R\left(n\right)$ takes on a fixed value $k$.

The goal of this paper is to outline the proof of a conjecture of Gelfond [6] (1968) in a recent work in collaboration with Christian Mauduit [11] concerning the sum of digits of prime numbers, reflecting the lecture given in Edinburgh at the Journées Arithmétiques 2007.

Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(n) is the number of integers in the interval [1,n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n)) < ϕ(n)...

Given an integer base $q\ge 2$ and a completely $q$-additive arithmetic function $f$ taking integer values, we deduce an asymptotic expression for the counting function$$N_f\left(x\right)=\#\left\{0\le n\&lt;x\left|f\right(n)\mid n\right\}$$under a mild restriction on the values of $f$. When $f=s_q$, the base $q$ sum of digits function, the integers counted by $N_f$ are the so-called base $q$ Niven numbers, and our result provides a generalization of the asymptotic known in that case.

For an integer $n\ge 2$ we denote by $P\left(n\right)$ the largest prime factor of $n$. We obtain several upper bounds on the number of solutions of congruences of the form $n!+2^n-1\equiv 0(modq)$ and use these bounds to show that$$\underset{n\to \infty }{lim\; sup}P(n!+2^n-1)/n\ge (2{\pi }^2+3)/18.$$

We provide upper bounds for the mean square integral$${\int }_X^{2X}{\left({𝔻}_k(x+h)-{𝔻}_k\left(x\right)\right)}^2\mathrm{d}x,$$where $h=h\left(X\right)\gg 1,h=o\left(x\right)\mathrm{as}X\to \infty$ and $h$ lies in a suitable range. For $k\ge 2$ a fixed integer, ${𝔻}_k\left(x\right)$ is the error term in the asymptotic formula for the summatory function of the divisor function $d_k\left(n\right)$, generated by ${\zeta }^k\left(s\right)$.