We denote by f(n, k) the number of positive integers m no,AC{1,2,...,n} with lAl>f(n,2) (if61n then f(n,2)= in), then the coprime graph induced by A contains a complete tripartite graph on 2 Lc,,,~~~;,,, J + 1 vertices where one of the classes ...

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...

Let Ak*(n) be the number of positive integers a coprime to n such that the equation a n=1 m1+ } } } +1 mk admits a solution in positive integers (m1 , ..., mk). We prove that the sum of A2*(n) over n x is both >>x log 3 x and also <<x log x. For the corresponding sum where the a's are counted with multiplicity of the number of solutions we obtain the asymptotic formula. We also show that Ak*(n)...

We classify primes p for which there exist elliptic curves E/Q with conductor NE ∈ {18p, 36p, 72p} and nontrivial rational 2-torsion, and, in consequence, show that, for “almost all” primes p, the Diophantine equation x + y = pz has at most finitely many solutions in coprime nonzero integers x, y and z and positive integers α and n ≥ 4. To prove this result, we appeal to such disparate techniqu...

Abstract Let (V,E) be the coprime graph with vertex set V = {1, 2, . . . , n} and edges (i, j) ∈ E if gcd(i, j) = 1. We determine the kernels of the coprime graph and its loopless counterpart as well as so-called simple bases for them (in case such bases exist), which means that basis vectors have entries only from {−1, 0, 1}. For the loopless version knowledge about the value distribution of M...

A. We use the group (Z2,+) and two associated automorphisms, τ0, τ1, to generate all distinct, non-zero pairs of coprime, positive integers which we describe within the context of a binary tree which we denote T . While this idea is related to the Stern-Brocot tree and the map of relatively prime pairs, the parents of an integer pair these trees do not necessarily correspond to the paren...

Let (ξ, y) be a point in R and ψ : N→ R a function. We investigate the problem of the existence of infinitely many pairs p, q of coprime integers such that |qξ + p− y| ≤ ψ(|q|). We give both unconditional results which are valid for every real pair (ξ, y) with ξ irrational, and metrical results valid for almost all points (ξ, y). We link the subject with density exponents of lattice orbits in R.

For any set of positive and relatively prime integers A, the set of positive integers that are not representable as a nonnegative integral linear combination of elements of A is always a non-empty finite set. Thus we may define g(A), n(A), s(A) to denote the largest integer in, the number of integers in, and the sum of integers in this finite set, respectively. We determine g(A), n(A), s(A) whe...

A P-set is a set S of positive integers such that no element of S divides the sum of any two (not necessarily being different) larger elements. Erdös and Sárközy [2] conjectured that there exists a constant c > 0 such that for every P-set S we have |{s ∈ S : s ≤ N}| < N1−c for infinitely many integers N . For P-sets S consisting of pairwise coprime integers this conjecture has been proved by T....

The discussion of Fractional dimensional Hilbert spaces in the context of Haldane exclusion statistics is extended from the case [10] of g = 1/p for the statistical parameter to the case of rational g = q/p with q, p-coprime positive integers. The corresponding statistical mechanics for a gas of such particles is constructed. This procedure is used to define the statistical mechanics for partic...