Notes on proof by dichotomy

Abstract

In this document we define a method of proof that we call proof by dichotomy. Its field of application is any proposition on the set of natural numbers N. It consists in the repetition of a step. A step proves the proposition for half of the members of an infinite subset U of N members for which we neither know if the proposition is verified nor not. We particularly study the case where the elements of U are separated by the parity of the quotient of euclidean division by 2 k. In such a case, we prove that if a natural n does not verify the proposition, then it is unique.