In this paper we study cycles in the coprime graph of integers. We denote by f(n, k) the number of positive integers m ≤ n with a prime factor among the first k primes. We show that there exists a constant c such that if A ⊂ {1, 2, . . . , n} with |A| > f(n, 2) (if 6|n then f(n, 2) = 2 3 n), then the coprime graph induced by A not only contains a triangle, but also a cycle of length 2l + 1 for ...

We nd innnitely many pairs of coprime integers, a and q, such that the least prime a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators.

New proof. Let n be an arbitrary positive integer greater than 1. Since n and n + 1 are consecutive integers, they must be coprime. Hence the number N2 = n(n + 1) must have at least two different prime factors. Similarly, since the integers n(n+1) and n(n+1)+1 are consecutive, and therefore coprime, the number N3 = n(n + 1)[n(n + 1) + 1] must have at least three different prime factors. This pr...

The integer d is called an exponential divisor of n = ∏r i=1 p ai i > 1 if d = ∏r i=1 p ci i , where ci|ai for every 1 ≤ i ≤ r. The integers n = ∏r i=1 p ai i ,m = ∏r i=1 p bi i > 1 having the same prime factors are called exponentially coprime if (ai, bi) = 1 for every 1 ≤ i ≤ r. In this paper we investigate asymptotic properties of certain arithmetic functions involving exponential divisors a...

The data in this article was obtained from the algebraic and statistical analysis of the first 331 primitive Pythagorean triples. The ordered sample is a subset of the larger Pythagorean triples. A primitive Pythagorean triple consists of three integers a, b and c such that; [Formula: see text]. A primitive Pythagorean triple is one which the greatest common divisor (gcd), that is; [Formula: se...

Let $L$ be a primitive Gaussian line, that is, line in the complex plane contains two, and hence infinitely many, coprime integers. We prove there exists an integer $G_L$ such for every $n\geq G_L$ are ma