Primes, coprimes and multiplicative elements

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

We find the formula for the cardinality of maximal set of integers from [1,. .. , n] which does not contain k + 1 pairwise coprimes and has divisors from a specified set of primes. This formula is defined by the set of multiples of the generating set, which does not depend on n. 1 Formulation of the result Let P = {p 1 < p 2 ,. . .} be the set of primes and N be the set of natural numbers. Writ...

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

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...

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...