On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a)

In this note, we study the representation of a natural number as the sum of three natural numbers having the form $\lfloor\frac{n^2}{a}\rfloor$ $(n \in \mathbb{N} )$ , where a is a fixed positive integer and $\lfloor . \rfloor$ denotes the integer-part function. By applying Gauss's triangular number theorem, we show that every natural number is the sum of three numbers of the form $\lfloor\frac{n^2}{8}\rfloor$ $(n \in \mathbb{N} )$ . And by applying Legendre's theorem, we show that every natural number is the sum of three numbers of the form $\lfloor\frac{n^2}{4}\rfloor$ $(n \in \mathbb{N} )$ and that every natural number N ≢ 2 (mod 24) is the sum of three numbers of the form $\lfloor\frac{n^2}{3}\rfloor$ $(n \in \mathbb{N} )$ . On the other hand, we show that every even natural number is the sum of three numbers of the form $\lfloor\frac{n^2}{2}\rfloor$ $(n \in \mathbb{N} )$ . We also propose two conjectures on the subject.

Received January 21 2013; revised versions received January 31 2013; May 25 2013; July 3 2013. Published in Journal of Integer Sequences, July 4 2013.

On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence $\lfloor\frac{n^2}{a}\rfloor$

Bakir Farhi
Department of Mathematics
University of Béjaia
Béjaia
Algeria

On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence

Bakir Farhi Department of Mathematics University of Béjaia Béjaia Algeria

On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence $\lfloor\frac{n^2}{a}\rfloor$

Bakir Farhi
Department of Mathematics
University of Béjaia
Béjaia
Algeria