Let a and b be positive integers and let $\{U_n\}_{n\ge 0}$ be the Lucas sequence of the first kind defined by

$$ \begin{align*}U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=aU_{n-1}+bU_{n-2} \quad \mbox{for }n\ge 2.\end{align*} $$

We define an $(a,b)$-Wall–Sun–Sun prime to be a prime p such that $\gcd (p,b)=1$ and $\pi (p^2)=\pi (p),$ where $\pi (p):=\pi _{(a,b)}(p)$ is the length of the period of $\{U_n\}_{n\ge 0}$ modulo p. When $(a,b)=(1,1)$, such primes are known in the literature simply as Wall–Sun–Sun primes. In this note, we provide necessary and sufficient conditions such that a prime p dividing $a^2+4b$ is an $(a,b)$-Wall–Sun–Sun prime.

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb {Z}[x]$ . We use an explicit version of Mertens’ theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb {Z}[x]$ , our result yields a finite list of primes that certifies the number of distinct irreducible factors of f.

Let $k\geqslant 1$ be a natural number and $\omega _k(n)$ denote the number of distinct prime factors of a natural number n with multiplicity k. We estimate the first and second moments of the functions $\omega _k$ with $k\geqslant 1$ . Moreover, we prove that the function $\omega _1(n)$ has normal order $\log \log n$ and the function $(\omega _1(n)-\log \log n)/\sqrt {\log \log n}$ has a normal distribution. Finally, we prove that the functions $\omega _k(n)$ with $k\geqslant 2$ do not have normal order $F(n)$ for any nondecreasing nonnegative function F.

For a polynomial $f(x)\in\mathbb{Q}[x]$ and rational numbers c, u, we put $f_c(x)\coloneqq f(x)+c$, and consider the Zsigmondy set $\calZ(f_c,u)$ associated to the sequence $\{f_c^n(u)-u\}_{n\geq 1}$, see Definition 1.1, where $f_c^n$ is the n-st iteration of fc. In this paper, we prove that if u is a rational critical point of f, then there exists an Mf > 0 such that $\mathbf M_f\geq \max_{c\in \mathbb{Q}}\{\#\calZ(f_c,u)\}$.

We prove irregularities in the distribution of prime numbers in any Beatty sequence ${\mathcal{B}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ is a positive real irrational number of finite type.

We show that under Dickson’s conjecture about the distribution of primes in the natural numbers, the theory Th (ℤ , +, 1, 0, Pr) where Pr is a predicate for the prime numbers and their negations is decidable, unstable, and supersimple. This is in contrast with Th (ℤ , +, 0, Pr, <) which is known to be undecidable by the works of Jockusch, Bateman, and Woods.

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.

We consider the sequences of fractional parts $\{\xi \alpha^n\}, n \,{=}\,1,2,3,\dots,$ and of integer parts $[\xi \alpha^n], n \,{=}\,1,2,3,\dots,$ where $\xi$ is an arbitrary positive number and $\alpha\,{>}\,1$ is an algebraic number. We obtain an inequality for the difference between the largest and the smallest limit points of the first sequence. Such an inequality was earlier known for rational $\alpha$ only. It is also shown that for roots of some irreducible trinomials the sequence of integer parts contains infinitely many numbers divisible by either 2 or 3. This is proved, for instance, for $[\xi((\sqrt {13}-1)/2)^n], n\,{=}\,1,2,3,\dots.$ The fact that there are infinitely many composite numbers in the sequence of integer parts of powers was proved earlier for Pisot numbers, Salem numbers and the three rational numbers 3/2, 4/3, 5/4, but no such algebraic number having several conjugates outside the unit circle was known.

We consider the primes which divide the denominator of the $x$-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive prime divisor, one that has not appeared in any earlier term. Proofs of this are known in only a few cases. Weaker results in the general direction are given, using a strong form of Siegel's Theorem and some congruence arguments. Our main result is applied to the study of prime divisors of Somos sequences.

We give a characterization of polynomials with rational coefficients which take integral values on the prime numbers: to test a polynomial of degree n, it is enough to consider its values on the integers from 1 to 2n —1.