Let $\alpha$ and $\beta$ are conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all $\alpha$ and $\beta$ with h$(\beta /\alpha )\le 4$, the $n$-th element of these sequences has a primitive divisor for $n\>30$. In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.

Let d be a fixed positive integer. A Lucas d-pseudoprime is a Lucas pseudoprime N for which there exists a Lucas sequence U(P,Q) such that the rank of appearance of N in U(P,Q) is exactly (N-ε(N))/d, where the signature ε(N) = (D/N) is given by the Jacobi symbol with respect to the discriminant D of U. A Lucas d-pseudoprime N is a primitive Lucas d-pseudoprime if (N-ε(N))/d is the maximal rank of N among Lucas sequences U(P,Q) that exhibit N as a Lucas pseudoprime. We derive...

Let $M\left(r\right)$ denote the set of real numbers $\alpha$ whose base-$r$ digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution. We show that the set $M\left(r\right)$ is closed under multiplication by rational numbers, but not closed under addition.

We give simple randomized algorithms leading to new upper bounds for combinatorial problems of Choi and Erdős: For an arbitrary additive group G let $P_n\left(G\right)$ denote the set of all subsets S of G with n elements having the property that 0 is not in S+S. Call a subset A of G admissible with respect to a set S from $P_n\left(G\right)$ if the sum of each pair of distinct elements of A lies outside S. Suppose first that S is a subset of the positive integers in the interval [2n,4n). Denote by f(S) the number of elements in a...

Let $\varphi (x_1,...,x_h,y)=u_1{x}_1+\cdots +u_h{x}_h+vy$ be a linear form with nonzero integer coefficients $u_1,...,u_h,v.$ Let $𝒜=(A_1,...,A_h)$ be an $h$-tuple of finite sets of integers and let $B$ be an infinite set of integers. Define the representation function associated to the form $\varphi$ and the sets $𝒜$ and $B$ as follows :$$R_{𝒜,B}^{\left(\varphi \right)}\left(n\right)=\text{card}\left(\begin{array}{cc}\hfill \{(a_1,...,a_h,b)\in A_1& \times \cdots \times A_h\times B:\hfill \\ & \varphi (a_1,...,a_h,b)=n\}\hfill \end{array}\right).$$If this representation function is constant, then the set $B$ is periodic and the period of $B$ will be bounded in terms of the diameter of the finite set $\{\varphi (a_1,...,a_h,0):(a_1,...,a_h)\in A_1\times \cdots \times A_h\}.$ Other results for complementing sets with respect to linear forms are also proved.

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$, then for every subset $B$ of $G$ with $\left|B\right|>\left|G\right|/k^{1/3}$ we have $B^3=G$. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k\ge 2$, then $G$ has a proper subgroup...