TY - JOUR

T1 - Unit fractions in norm-Euclidean rings of integers

AU - Bradford, K.

AU - Ionascu, E. J.

N1 - Publisher Copyright:
© 2017, Univerzita Komenskeho. All rights reserved.

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - Diophantine equations

KW - Integers

UR - http://www.scopus.com/inward/record.url?scp=85009469027&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85009469027

SN - 0862-9544

VL - 86

SP - 127

EP - 141

JO - Acta Mathematica Universitatis Comenianae

JF - Acta Mathematica Universitatis Comenianae

IS - 1

ER -