On representations of real quadratic integers as sums of unitary fractions

Germán Benitez; Elkin Quintero Vanegas; Euro Vidal Sampaio

doi:10.5269/bspm.67322

On representations of real quadratic integers as sums of unitary fractions

Germán Benitez Universidade Federal do Amazonas https://orcid.org/0000-0003-0565-6300
Elkin Quintero Vanegas Universidade Federal do Amazonas - UFAM https://orcid.org/0000-0002-6641-0470
Euro Vidal Sampaio Universidade Federal do Amazonas

Resumo

We verify that each natural number $t>1$ can be expressed as a sum of unitary fractions of different natural numbers such that the least common multiple among them is coprime with a fixed value $v$. Using this fact, we characterize the real quadratic integers which appear in the representation of 1 as a unitary fraction sum over the real quadratic integer ring.