arXiv subjects

Laurent Fallot

Publications and source records attributed to Laurent Fallot.

2 recordsLinked to original sources

arXiv2023

The monoid of numbers of the form 1 < a^q /b^p < a

This paper is a study of the set of rational numbers of the form 1 < a^q /b^p < a with a and b co-prime integers. The set F (a,b) of these numbers, with an appropriate binary law, is a monoid isomorphic to (N, +, 0). We identify the sequences of minimum and maximum record holders in F (a,b) and prove that the first one converges to 1 while the second one converges to a. We conclude that F (a,b) is dense in the set of the real numbers comprise between 1 and a.

math.GN↗

arXiv2023

Notes on proof by dichotomy

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.

math.LO↗