Rational θ-parallelogram envelopes via θ-congruent elliptic curves

2020/12/25 by Sajad Salami, Salami, Sajad, Arman Shamsi Zargar +1
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry and Number Theory #History and Theory of Mathematics #math.NT #msc:11G05 #msc:14H52

paper · pdf · doi:10.48550/arxiv.2012.13471

21 pages

arxiv created 2021/03/30 · arxiv updated 2021/03/31

Abstract

We introduce a new generalization of θ-congruent numbers by defining the notion of rational θ-parallelogram envelope for a positive integer n, where θ∈ (0, π) is an angle with rational cosine. Then, we study more closely some problems related to the rational θ-parallelogram envelopes, using the arithmetic of algebraic curves. Our results generalize the recent work of T.~Ochiai, where only the case θ=π/2 was considered. Moreover, we answer the open questions in his paper and their generalizations for any Pythagorean angle.

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