Elementary results on the binary quadratic form a^2+ab+b^2

Abstract

This paper examines with elementary proofs some interesting properties of numbers in the binary quadratic form $a^2+ab+b^2$, where $a$ and $b$ are non-negative integers. Key findings of this paper are (i) a prime number $p$ can be represented as $a^2+ab+b^2$ if and only if $p$ is of the form $6k+1$, with the only exception of 3, (ii) any positive integer can be represented as $a^2+ab+b^2$ if and only if its all prime factors that are not in the same form have even exponents in the standard factorization, and (iii) all the factors of an integer in the form $a^2+ab+b^2$, where $a$ and $b$ are positive and relatively prime to each other, are also of the same form. A general formula for the number of distinct representations of any positive integer in this form is conjectured. A comparison of the results with the properties of some other binary quadratic forms is given.