On additive decompositions of the set of primes

We show that there is no set A {\cal A} of integers, such that¶ (P - 1) \subseteqq A + A \subseteqq P è(P - 1) ({\cal P} - {1}) \subseteqq {\cal A} + {\cal A} \subseteqq {\cal P} \cup ({\cal P} - 1) ,¶ where P {\cal P} denotes the set of primes.     

On the least prime primitive root modulo a prime

We derive a conditional formula for the natural density of prime numbers having its least prime primitive root equal to , and compare theoretical results with the numerical evidence.

     

A remark on Hofmann and Wolke's additive decompositions of the set of primes

We improve upon Hofmann and Wolke's bounds on the conceivable additive decompositions of the set of primes. We also prove that there cannot be a decomposition, where one set is a shifted copy of the other. This answers an open question of Hofmann and Wolke.     

On functional which are orthogonally additive modulo Z

Let E be a real inner product space with dimension at least 2, D ? E, f: E → R with f(x+y)?f(x)?f(y) ∈ Z for all orthogonal x,y ∈ E, and f(D) ? (?γ,γ)+Z witn some real γ > 0. We prove that, under some additional assumptions, there are a unique linear functional A: E → R and a unique constant d ∈ R with f(x)?d∥x∥²?A(x) ∈ Z for x ∈ E. We also show some applications of this result to the determination of solutions F: E → C of the conditional equation: F(x+y) = F(x)F(y) for all orthogonal x,y ∈ E.     

Group algebras primitive modulo the Jacobson radical

Let K[G] denote the group algebra of the group G over the field K. Also let J(K[G]) be the Jacobson radical. The main aim in this paper is to study the following question: When is K[G]/J(K[G]) primitive? We mainly investigate semilocal group algebras, group algebras satisfying a polynomial identity and group algebras of a suitable direct product and that of a polycyclic-by-finite group. Many examples of interest and some elementary ring theoretic properties are also given.     

On primitive roots for Carlitz modules

Let be the polynomial ring over a finite field. We prove that for every element a of a global -field of finite -characteristic the set of places for which a is a primitive root under the Carlitz action possesses a Dirichlet density. We also give a criterion for this density to be positive. This is an analogue of Bilharz’ version of the primitive roots conjecture of Artin, with replaced by the Carlitz module.     

On convex decompositions of a planar point set

Let P be a planar point set in general position. Neumann-Lara et al. showed that there is a convex decomposition of P with at most elements. In this paper, we improve this upper bound to .     

On the distribution of square-full and cube-full primitive roots

Periodica Mathematica Hungarica - A positive integer n is called an r-full integer if for all primes $$p\mid n$$ we have $$p^r\mid n.$$ Let p be an odd prime. For $$\gcd (n,p)=1$$, the smallest...     

On the pseudo-randomness of subsets related to primitive roots

Many results have been proved on the distribution of the primitive roots. These results reflect certain random type properties of the set G _p of the primitive roots modulo p. This fact motivates the question that in what extent behaves G _p as a random subset of ? _p ? First a much more general form of this problem is studied by using the notion of pseudo-randomness of subsets of ? _n which has been introduced and studied recently by Dartyge and Sárközy. This is followed by the study of the pseudo-randomness of a subset of ? _p defined by index properties. In both cases it turns out that these subsets possess strong pseudo-random properties (the well-distribution measure and correlation measure of order k are small) but the pseudo-randomness is not perfect: there is a pseudo-random measure (the symmetry measure) which is large.     

On the distribution of the primitive roots of a prime

In this paper it is shown that the number of pairs of consecutive primitive roots modulo p is asymptotic to $\text{(p ? 2)(}\text{?(p ? 1)}\text{(p ? 1)}\text{)}{}^2$, and that, for all sufficiently large primes p, there is at least one pair of consecutive primitive roots modulo p. The theorem proved here is a generalization of this proposition. Another one is mentioned in the remarks.     

ON THE EXPONENT SET OF PRIMITIVELOCALLY SEMICOMPLETE DIGRAPHS

A locally semicomplete digraph is a digraph D=(V,A) satisfying the following condi-tion for every vertex x∈V the D[O(x)] and D[I(x)] are semicomplete digraphs. In this paper,we get some properties of cycles and determine the exponent set of primitive locally semicompleted digraphs.     

On a conjecture of rodier on primitive roots

On the Generalized Riemann Hypothesis (GRH) a conjecture of Rodier, on the set of primesp such that 2 is a primitive root modp, is disproved.     

On the set of orthogonally additive functions with orthogonally additive second iterate

Let E be a real inner product space of dimension at least 2. We show that both the set of all orthogonally additive functions mapping E into E having orthogonally additive second iterate and its complement are dense in the space of all orthogonally additive functions from E into E with the Tychonoff topology.     

On primitive roots for rank one Drinfeld modules

Let k be a global function field over a finite field and let A be the ring of the elements in k regular outside a fixed place ∞. Let K be a global A-field of finite A-characteristic and let ? be a rank one Drinfeld A-module over K. Given any α∈K, we show that the set of places P of K for which α is a primitive root modulo P under the action of ? possesses a Dirichlet density. We also give conditions for this density to be positive.     

Hamilton decompositions of graphs with primitive complements

A k-factor of a graph is a k-regular spanning subgraph. A Hamilton cycle is a connected 2-factor. A graph G is said to be primitive if it contains no k-factor with 1≤k<Δ(G). A Hamilton decomposition of a graph G is a partition of the edges of G into sets, each of which induces a Hamilton cycle. In this paper, by using the amalgamation technique, we find necessary and sufficient conditions for the existence of a 2x-regular graph G on n vertices which:

1.

has a Hamilton decomposition, and

2.

has a complement in K_n that is primitive.

This extends the conditions studied by Hoffman, Rodger, and Rosa [D.G. Hoffman, C.A. Rodger, A. Rosa, Maximal sets of 2-factors and Hamiltonian cycles, J. Combin. Theory Ser. B 57 (1) (1993) 69-76] who considered maximal sets of Hamilton cycles and 2-factors. It also sheds light on construction approaches to the Hamilton-Waterloo problem.