Notes on proof by dichotomy

TL;DR

This paper introduces a proof method called proof by dichotomy applicable to propositions on natural numbers, involving iterative steps that halve the problem set, with a focus on cases distinguished by parity and quotient properties.

Contribution

It formalizes the proof by dichotomy method and analyzes its application, especially in cases based on parity and Euclidean division, providing new insights into proof uniqueness.

Findings

Proof by dichotomy applies to propositions on natural numbers.

In certain cases, non-verifying elements are unique.

The method involves iterative halving of the problem set.

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.