Andrew Bremner, Samir Siksek
Published 2015-05-24Version 1
Euler showed that there can be no more than three integer squares in arithmetic progression. In quadratic number fields, Xarles has shown that there can be arithmetic progressions of five squares, but not of six. Here, we prove that there are no cubic number fields which contain five squares in arithmetic progression.
Three cubes in arithmetic progression over quadratic fields
Distributions of Arithmetic Progressions in Piatetski-Shapiro Sequence
When the Nontrivial, Small Divisors of a Natural Number are in Arithmetic Progression
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