Let $P(n)$ denote the largest prime factor of an integer $n\geq 2$. In this paper, we study the distribution of the sequence $\{f(P(n)):n\geq 1\}$ over the set of congruence classes modulo an integer $b\geq 2$, where $f$ is a strongly $q$-additive integer-valued function (that is, $f(aq^{j}+b)=f(a)+f(b),$ with $(a,b,j)\in \mathbb{N}^{3}$, $0\leq b<q^{j}$). We also show that the sequence $\{{\it\alpha}P(n):n\geq 1,f(P(n))\equiv a\;(\text{mod}~b)\}$ is uniformly distributed modulo 1 if and only if ${\it\alpha}\in \mathbb{R}\!\setminus \!\mathbb{Q}$.

Given an integer $q\ge 2$, a $q$-normal number is an irrational number $\eta $ such that any preassigned sequence of $\ell $ digits occurs in the $q$-ary expansion of $\eta $ at the expected frequency, namely $1/q^\ell $. In a recent paper we constructed a large family of normal numbers, showing in particular that, if $P(n)$ stands for the largest prime factor of $n$, then the number $0.P(2)P(3)P(4)\ldots ,$ the concatenation of the numbers $P(2), P(3), P(4), \ldots ,$ each represented in base $q$, is a $q$-normal number, thereby answering in the affirmative a question raised by Igor Shparlinski. We also showed that $0.P(2+1)P(3+1)P(5+1) \ldots P(p+1)\ldots ,$ where $p$ runs through the sequence of primes, is a $q$-normal number. Here, we show that, given any fixed integer $k\ge 2$, the numbers $0.P_k(2)P_k(3)P_k(4)\ldots $ and $0. P_k(2+1)P_k(3+1)P_k(5+1) \ldots P_k(p+1)\ldots ,$ where $P_k(n)$ stands for the $k{\rm th}$ largest prime factor of $n$, are $q$-normal numbers. These results are part of more general statements.

Given an integer d≥2, a d-normal number, or simply a normal number, is an irrational number whosed-ary expansion is such that any preassigned sequence, of length k≥1, taken within this expansion occurs at the expected limiting frequency, namely 1/dk. Answering questions raised by Igor Shparlinski, we show that 0.P(2)P(3)P(4)…P(n)… and 0.P(2+1)P(3+1)P(5+1)…P(p+1)…, where P(n) stands for the largest prime factor of n, are both normal numbers.

The largest prime factor of $X^{3}+2$ was investigated in 1978 by Hooley, who gave a conditional proof that it is infinitely often at least as large as $X^{1+\delta}$, with a certain positive constant $\delta$. It is trivial to obtain such a result with $\delta=0$. One may think of Hooley's result as an approximation to the conjecture that $X^{3}+2$ is infinitely often prime. The condition required by Hooley, his R$^{*}$ conjecture, gives a non-trivial bound for short Ramanujan--Kloosterman sums. The present paper gives an unconditional proof that the largest prime factor of $X^{3}+2$ is infinitely often at least as large as $X^{1+\delta}$, though with a much smaller constant than that obtained by Hooley. In order to do this we prove a non-trivial bound for short Ramanujan--Kloosterman sums with smooth modulus. It is also necessary to modify the Chebychev method, as used by Hooley, so as to ensure that the sums that occur do indeed have a sufficiently smooth modulus. 2000 Mathematics Subject Classification: 11N32.