Unit fractions in norm-Euclidean rings of integers

In this article, we consider the Erdős-Straus conjecture in a more general setting. For instance, one can look at the diophantine equation (Forumal presented) where n and a, b, c are Gaussian integers. We have considered this problem in the case of rings of integers of the norm-Euclidean quadratic fields. Without any other restrictions on a, b and c, we show that solutions exist except for a finite set, which is given explicitly in each particular case. The problem becomes as difficult as the original Erdős-Straus conjecture if we require that all variables are in the first or third quadrant, but numerical evidence shows a decomposition still exists. We formulate this new conjecture explicitly in the end of this article.