Testability of positive integers

Abstract

Given a set [Formula: see text] of some positive integers, a positive integer [Formula: see text] is [Formula: see text]-testable if there are two disjoint subsets [Formula: see text] and [Formula: see text] of [Formula: see text] such that [Formula: see text]. The testability of the set [Formula: see text] of positive integers not exceeding [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that there is a set [Formula: see text] with [Formula: see text] positive integers satisfying that each [Formula: see text] is [Formula: see text]-testable. In this short paper, we prove that [Formula: see text] and propose an algorithm to output the representation of an arbitrary positive integer by the form [Formula: see text], where [Formula: see text] and [Formula: see text] are two disjoint sets of some non-negative integers not exceeding [Formula: see text].