On a Conjecture on the Representation of Positive Integers as the Sum of Three Terms of the Sequence $\left\lfloor \frac{n^2}{a} \right\rfloor$

TL;DR

This paper proves certain cases of a conjecture by Farhi, showing that every positive integer can be expressed as the sum of three terms from the sequence loor(n^2/a), by generalizing Farhi's method.

Contribution

It extends Farhi's work by proving new cases of the conjecture through a generalized approach.

Findings

Confirmed some cases of the conjecture

Generalized the method used by Farhi

Enhanced understanding of the sequence's additive properties

Abstract

We prove some cases of a conjecture by Farhi on the representation of every positive integer as the sum of three terms of the sequence $ \left\lfloor\frac{n^2}{a}\right\rfloor$. This is done by generalizing a method used by Farhi in his original paper.