The delta-lambda configurations in tiling the square
type: Item
instance of: scholarly article
title: The delta-lambda configurations in tiling the square
zbMATH Open document ID: 0678.05013
DOI: 10.1016/0022-314X(89)90083-8
author: Andrew Bremner
author: Richard K. Guy
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: The authors begin with the problem of tiling an integer sided square with integer sided triangles. There are essentially four possible configurations consisting of four triangles, which they call the nu, delta, chi, and kappa configurations. The main purpose of this note is a fairly thorough discussion of the delta configuration. A number x is called a rectangular number if \(x^ 2+1\) is a perfect square. The authors show that the delta configurations are characterized by solutions of the equation \(1-Y=M(1-X)\) with X,Y,M all rectangular numbers. This quickly leads to the problem of finding all rational points on a certain surface. After several transformations, the authors exhibit this surface (birationally) as an elliptic surface E over \({\mathbb{P}}^ 1\) given by the Weierstrass equation \[ y^ 2=x[x+2(\lambda +1)^ 2][x+2\lambda^ 2(\lambda -1)^ 2]. \] They perform a standard 2-descent to show that E has rank 1 over \({\mathbb{Q}}(\lambda)\). (However, they leave open the question of the rank of E over (\({\mathbb{C}}(\lambda).)\). This leads to an infinite family of parametrized solutions to the delta configuration. Next they substitute specific rational values for \(\lambda\). If the resulting elliptic curve has rank 1 over \({\mathbb{Q}}\), they essentially obtain all delta configurations for that \(\lambda\). However, it can happen that the rank jumps. They perform a descent to find an upper bound for the rank. Finally, they find many values of \(\lambda\) for which the rank is 2, including a certain one-parameter family, and also one value for which the rank is 3.
Mathematics Subject Classification ID: 05B45
Mathematics Subject Classification ID: 11D04
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 4110706
zbMATH Keywords: tiling
zbMATH Keywords: rectangular number
zbMATH Keywords: elliptic curve
MaRDI profile type: Publication
cites work: Notes on elliptic curves. II.
cites work: Diophantine Equations with Special Reference To Elliptic Curves
cites work: On sums of squares and on elliptic curves over function fields
cites work: Q3977456
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cites work: Q5582375
description: scientific article; zbMATH DE number 4110706