Equilateral triangles in Z⁴

2012/09/02 by Eugen J. Ionaşcu, Eugen J. Ionascu, Ionascu, Eugen J.
Engineering · Mathematics · #11B99 #11D04 #11R99 #Analytic Number Theory Research #Diverse Scientific and Engineering Research #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT) #Primary 11A67 #Secondary 11D09 #math.NT #msc:11A67 #msc:11B99 #msc:11D04 #msc:11D09 #msc:11R99

paper · pdf · doi:10.48550/arxiv.1209.0147

1 figure

openalex publication_date 2012/09/02 · arxiv created 2013/07/14 · arxiv updated 2013/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a characterization of all three points in \mathbb R⁴ with integer coordinates which are at the same Euclidean distance apart. In three dimension the problem is characterized in terms of solutions of the Diophantine equations a²+b²+c²=3d². In \mathbb R⁴, the characterization is essentially based on two different solutions of the same equation.

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