Our purpose is to establish the following result: Let a and d be coprime integers and a, a+ d, a+ 2d, . . . , a+ (k − 1) d (k > 2) be an arithmetical progression. Then for all integers α0, α1, . . . , αk−1 the rational number 1/a0 + 1/ (a + d)α1 + · · · + 1/ (a + (k − 1) d)αk−1 is never an integer. This result extends theorems of Taeisinger (1915) and Kürschák (1918), and also generalizes a res...

In the Frobenius problem we are given a set of coprime, positive integers a1, a2, . . . , ak, and are interested in the set of positive numbers NR that have no representation by the linear form ∑ i aixi in nonnegative integers x1, x2, . . . , xk. We give a functional relationship that completely characterizes the set NR, and apply it to the case when the numbers are in an arithmetic progression.

Let a1, a2, . . . , ak be positive and pairwise coprime integers with product P . For each i, 1 ≤ i ≤ k, set Ai = P/ai. We find closed form expressions for the functions g(A1, A2, . . . , Ak) and n(A1, A2, . . . , Ak) that denote the largest (respectively, the number of) N such that the equation A1x1 + A2x2 + · · · + Akxk = N has no solution in nonnegative integers xi. This is a special case of...

In the field of mathematics, there are many branches study, especially in graph theory, mathematically a is pair sets, which consists non-empty set whose members called vertices and distinct unordered pairs edges. One example from group coprime graph, where defined as two with different x y neighbors if only (|x|,|y|)=1. this author discusses Harmonic Index Gutman Coprime Graph Integer Group Mo...

Fix pairwise coprime positive integers p1, p2, . . . , ps. We propose representing integers u modulo m, where m is any positive integer up to roughly √ p1p2 · · · ps, as vectors (u mod p1, u mod p2, . . . , u mod ps). We use this representation to obtain a new result on the parallel complexity of modular exponentiation: there is an algorithm for the Common CRCW PRAM that, given positive integer...

A set A of positive integers is called a coprime Diophantine powerset if the shifted product ab + 1 of two different elements a and b of A is always a pure power, and the occurring pure powers are all coprime. We prove that each coprime Diophantine powerset A ⊂ {1, . . . , N} has |A| 8000 log N/ log log N for sufficiently large N. The proof combines results from extremal graph theory with numbe...

We construct a Kirby diagram of the rational homology ball used in “generalized rational blow-down” developed by Jongil Park. The diagram consists of a dotted circle and a torus knot. The link is simpler, but the parameters are a little complicate. Euclidean Algorithm is used three times in the construction and the proof. 1 Main theorem For a coprime pair (m,n) of positive integers, we take a s...