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

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 sequences containing at most 4 pairwise coprime integers

It has been conjectured by Erdó's that the largest number of natural numbers not exceeding n from which one cannot select k + l pairwise coprime integers, where k > 1 and n > p,, with p, denoting the zcth prime, is equal to the number of natural numbers not exceeding n which are multiples of at least one of the first k primes. It is known that the conjecture holds for k = 1 and 2. In this paper...

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...

Proof of D. J. Newman's Coprime Mapping Conjecture

We call the function / described in the theorem a coprime mapping. Theorem 1 settles in the affirmative a conjecture of D. J. Newman. The special case when / = {JV + l.JV + 2, ...,2Af} was proved by D. E. Daykin and M. J. Baines [2]. V. Chvatal [1] established Newman's conjecture for each JV ^ 1002. We prove Theorem 1 constructively by giving an algorithm for the construction of a coprime mappi...

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