An Application of Graph Theory to Additive Number Theory

An Application of Graph Theory to Additive Number Theory

A sequence of integers A = {a, < a2 < • • . < a„} is a Bzk) sequence if the number of representations of every integer as the sum of two distinct a,s is at most k . In this note we show that every B' sequence of n terms is a union of C2(k) • n 113 BzB (1) sequences, a nd tatthe re is aB (k) sequence of n terms which is not a union of cik • n'" 3 Bz" sequences. This solves a problem raised in 2 2

An asymptotic formula in additive number theory

1 . Introduction . In his paper [1], Erdös introduced the sequences of positive integers b 1 < b, < . . ., with (b ;, bj ) = 1, for i ~A j, and 'bi 1 < oo . With any such arbitrary sequence of integers, he associated the sequence {di} of all positive integers not divisible by any bj , and he showed that if b1 > 2, there exists a 0 < 1 (independent of the sequence {b i }) such that d i 1 di < d°...

Ergodic Theory and Applications to Additive Number Theory

The following lemma follows the standard paradigm of recurrence results in ergodic theory: given a topological space X which satisfies a suitable ’smallness’ condition (e.g. compactness, μ(X) < ∞), and a transformation T : X → X, there exist points of X which satisfy a certain ’almost-periodicity’ condition under the action of T . Lemma 1. Poincare Recurrence Lemma If (X,β, μ, T ) is an m.p.s. ...

Application of Graph Theory to Some Thermodynamic Properties and Topological Indices

The relationship between the Randic , Wiener, Hosoya , Balaban, Schultz indices, Harary numbers andDistance matrix to enthalpies of formation (Airf), heat capacity, (Cp) , enthalpies of combustion (AH °c ),enthalpy of vaporization (AH °vap) and normal boiling points (bpK)of C2 C10 normal alkanes isrepresented

On Application of Graph Theory to Networks and Coding Theory