Unit Fractions in Norm-Euclidean Rings of Integers

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Eugen J. Ionascu Kyle Bradford


In this article we consider the Erdos-Straus conjecture in a more general setting. For instance, one can look at the diophantine equation                 4/n = 1/a + 1/b + 1/cwhere n and a; b; c are Gaussian integers. The problem becomes as difficult as the original Erdos-Straus conjecture if we require that the solutions be in the first or third quadrant for every given n in the first quadrant different of 0, 1, i, or 1 + i. However, without any other restrictions on a, b and c, we show that solutions exist except for a finite set. We considered also the caseof rings of integers of the norm-Euclidean quadratic fields.

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Ionascu, E., & Bradford, K. (2017). Unit Fractions in Norm-Euclidean Rings of Integers. Acta Mathematica Universitatis Comenianae, 86(1), 127-141. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/368/420