Chords of an ellipse, Lucas polynomials, and cubic equations

• Published in 2018
• Added on 2019-06-30

A beautiful result of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse. We give a conceptually transparent development of this result that provides a tour of several gems of classical mathematics: It is inspired by Girolamo Cardano's solution of the cubic equation, uses Newton's theorem connecting power sums and elementary symmetric polynomials, and yields for free an alternative proof of the Binet formula for the generalized Lucas polynomials.

Links

• http://arxiv.org/abs/1810.00492v3
• http://arxiv.org/pdf/1810.00492v3

Other information

BibTeX entry

@article{ChordsofanellipseLucaspolynomialsandcubicequations,
	key = {ChordsofanellipseLucaspolynomialsandcubicequations},
	type = {article},
	title = {Chords of an ellipse, Lucas polynomials, and cubic equations},
	author = {Ben Blum-Smith and Japheth Wood},
	abstract = {A beautiful result of Thomas Price links the Fibonacci numbers and the Lucas
polynomials to the plane geometry of an ellipse. We give a conceptually
transparent development of this result that provides a tour of several gems of
classical mathematics: It is inspired by Girolamo Cardano's solution of the
cubic equation, uses Newton's theorem connecting power sums and elementary
symmetric polynomials, and yields for free an alternative proof of the Binet
formula for the generalized Lucas polynomials.},
	comment = {},
	date_added = {2019-06-30},
	date_published = {2018-12-07},
	urls = {http://arxiv.org/abs/1810.00492v3,http://arxiv.org/pdf/1810.00492v3},
	collections = {Easily explained,Fun maths facts,Puzzles},
	url = {http://arxiv.org/abs/1810.00492v3 http://arxiv.org/pdf/1810.00492v3},
	year = 2018,
	urldate = {2019-06-30},
	archivePrefix = {arXiv},
	eprint = {1810.00492},
	primaryClass = {math.HO}
}