θ-Congruent Numbers, Tiling Numbers and the Selmer Rank of Related Elliptic Curves: odd n

2020/10/19 by Liu, Qiuyue, Yang, Jing, Feng, Keqin
#FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2010.09238

Abstract

Several discrete geometry problems are closely related to the arithmetic theory of elliptic curves defined on the rational fields ℚ. In this paper we consider the θ-congruent number for θ=\fracπ3 and (2π)/(3) and tiling number n. For the case that n\geqslant 2 is square-free odd integer, we determine all n such that the Selmer rank of elliptic curve E_n,\fracπ3: y²=x(x-n)(x+3n) or/and E_n,(2π)/(3): y²=x(x+n)(x-3n) is zero. From this, we provide several series of non θ-congruent numbers for θ=\fracπ3 and (2π)/(3), and non tiling numbers n with arbitrary many of prime divisors.

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