Squares in arithmetic progression over cubic fields

2015/05/24 by Andrew Bremner, Samir Siksek, Bremner, Andrew +1
Mathematics · #11B25 #11G30 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #math.NT #msc:11B25 #msc:11G30

paper · pdf · doi:10.48550/arxiv.1505.06424

arxiv created 2015/05/24 · openalex publication_date 2015/05/24 · arxiv updated 2015/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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