Every Real Algebraic Integer Is a Difference of Two Mahler Measures

We prove that every real algebraic integer $\alpha$ is expressible by a difference of two Mahler measures of integer polynomials. Moreover, these polynomials can be chosen in such a way that they both have the same degree as that of $\alpha$ , say $d$ , one of these two polynomials is irreducible and another has an irreducible factor of degree $d$ , so that $\alpha =M\left( P \right)-bM\left( Q \right)$ with irreducible polynomials $P,Q\in \mathbb{Z}\left[ X \right]$ of degree $d$ and a positive integer $b$ . Finally, if $d\le 3$ , then one can take $b=1$ .