On Representable Numbers and a Conjecture by Erdos and Lewin

Final Defense
On Representable Numbers and a Conjecture by Erdos and Lewin

by Minchan Jeong
MS Mathematics Candidate

Date: Monday, 29 April 2024
Time: 3:30 pm
Venue: SEC A 321 (MJR Room)

Advisers:
Mark Anthony A. Tolentino, PhD
Ateneo de Manila University

Panelists:
Mark L. Loyola, PhD
Ateneo de Manila University
Job A. Nable, PhD
Ateneo de Manila University
Winfer C. Tabares
Ateneo de Manila University

A positive integer n is said to be a representable number (with respect to positive integers p and q) if n can be expressed as a sum of integers of the form pα qβ , with no summand dividing any other summand. This paper aims to provide an exposition on a conjecture by Erdos and Lewin, representable numbers, and some related results. We begin by expanding on the original work of Erdos and Lewin who has established, among other results, that every positive integer is representable with respect to 2 and 3. We then discuss a recent work of Yu and Chen who have partially answered Erdos’ and Lewin’s question on the density of numbers repre- sentable with respect to p and q, where {p,q} ̸= {2,3}, and who have proven a conjecture on representability using bounded addends. Finally, we also provide a construction of representable numbers satisfying a pre- scribed divisibility condition.

Key Words: Representable Numbers, Erdos-Lewin’s conjecture, d-complete sequences, Divisibility