Let a and f be coprime positive integers. Let g be an integer. Under the Generalized Riemann Hypothesis (GRH) it follows by a result of H.W. Lenstra that the set of primes p such that p ≡ a(mod f) and g is a primitive root modulo p has a natural density. In this note this density is explicitly evaluated with an Euler product as result.

Let n and k be integers. A set A ⊂ Z/nZ is k-free if for all x in A, kx / ∈ A. We determine the maximal cardinality of such a set when k and n are coprime. We also study several particular cases and we propose an efficient algorithm for solving the general case. We finally give the asymptotic behaviour of the minimal size of a k-free set in J1, nK which is maximal for inclusion.

The coprime graph of a group G is Γ_G with its set vertices and any two distinct are adjacent if only their order relatively prime. Let p be prime number, then G_p denotes the multiplicative 2×2 upper unitriangular matrices over ring integers modulo p. purposes this research to study Γ_(G_p ) find first second Zagreb eccentricity indices for p≥3. results as follows. First index )isE_1 (Γ_(G_p )...

In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime [Formula: see text] where is the text]th number. This was generalized by Wang Cai in 2014, Cai, Shen Jia 2017 replacing summation modulus with an power, product of two powers, respectively. particular, proposed conjectural congruence: positive integer factor such that this paper, we ...

On a regular tetrahedron in spherical space there exist the finite number of simple closed geodesics. For any pair coprime integers $(p,q)$ it was found numbers $\alpha_1$ and $\alpha_2$ depending on $p$, $q$ satisfying inequalities $\pi/3< \alpha_1 < \alpha_2 2\pi/3$ such that with faces angle $\alpha \in \left( \pi/3, \right)$ exists unique, up to rigid motion tetrahedron, geodesic type $(p,q...

Let n be a positive integer and αn be the arithmetic function which assigns the multiplicative order of an modulo n to every integer a coprime to n and vanishes elsewhere. Similarly, let βn assign the projective multiplicative order of a n modulo n to every integer a coprime to n and vanish elsewhere. In this paper, we present a study of these two arithmetic functions. In particular, we prove t...

We study the behavior of the greatest common divisor of a k − 1 and b k − 1, where a, b are fixed integers or polynomials, and k varies. In the integer case, we conjecture that when a and b are multiplicatively independent and in addition a− 1 and b − 1 are coprime, then a k − 1 and b k − 1 are coprime infinitely often. In the polynomial case, we prove a strong version of this conjecture. To do...

We study the random Fibonacci tree, which is an infinite binary tree with non-negative numbers at each node defined as follows. The root consists of the number 1 with a single child also the number 1. Then we define the tree recursively in the following way: if x is the parent of y, then y has two children, namely |x−y| and x+y. This tree was studied by Benoit Rittaud [?] who proved that any pa...

Paul Erdos defined the concept of coprime graph and studied about cycles in coprime graphs. In this paper this concept is generalized and a new graph called Generalized coprime graph is introduced. Having observed certain basic properties of the new graph it is proved that the chromatic number and the clique number of some generalized coprime graphs are equal.

We prove that any set of integers A ⊂ [1, x] with |A| (log x)r lies in at least νA(p) p r r+1 many residue classes modulo most primes p (log x)r+1. (Here r is a positive constant.) This generalizes a result of Erdős and Ram Murty, who proved in connection with Artin’s conjecture on primitive roots that the integers below x which are multiplicatively generated by the coprime integers a1, . . . ,...