Maximal sets of integers not containing k + 1 pairwise coprimes and having divisors from a specified set of primes

Maximal sets of numbers not containing k+1 pairwise coprimes and having divisors from a specified set of primes

Maximal sets of numbers not containing k + 1 pairwise coprime integers

For positive integers k; n let f(n; k) be the maximal cardinality of subsets of integers in the interval < 1; n > , which don't have k + 1 pairwise coprimes. The set E (n; k) of integers in < 1; n > , which are divisible by one of the rst k primes, certainly does not have k + 1 pairwise coprimes. Whereas we disproved in [1] an old conjecture of Erdos ([4], [5], [6], [7]) by showing that the eq...

Maximal sets of numbers not containing k + 1 pairwise coprime

For positive integers k, n let f(n, k) be the maximal cardinality of subsets of integers in the interval < 1, n > , which don’t have k + 1 pairwise coprimes. The set E(n, k) of integers in < 1, n > , which are divisible by one of the first k primes, certainly does not have k + 1 pairwise coprimes. Whereas we disproved in [1] an old conjecture of Erdös ([4], [5], [6], [7]) by showing that the eq...

On Extremal Sets without Coprimes

1. Definitions, formulation of problems and conjectures. We use the following notations: Z denotes the set of all integers, N denotes the set of positive integers, and P = {p 1 , p 2 ,. . .} = {2, 3, 5,. . .} denotes the set of all primes. We set (1.1) Q k = k i=1 p i. For two numbers u, v ∈ N we write (u, v) = 1 if u and v are coprimes. We are particularly interested in the sets (1.2) N s = {u...

The power digraphs of safe primes

A power digraph, denoted by $G(n,k)$, is a directed graph with $Z_{n}={0,1,..., n-1}$ as the set of vertices and $L={(x,y):x^{k}equiv y~(bmod , n)}$ as the edge set, where $n$ and $k$ are any positive integers. In this paper, the structure of $G(2q+1,k)$, where $q$ is a Sophie Germain prime is investigated. The primality tests for the integers of the form $n=2q+1$ are established in terms of th...