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October 2020 On the area bounded by the curve $\prod_{k = 1}^n\left|x\sin\frac{k\pi}{n} - y\cos\frac{k\pi}{n}\right|=1$
Anton Mosunov
Rocky Mountain J. Math. 50(5): 1773-1777 (October 2020). DOI: 10.1216/rmj.2020.50.1773

Abstract

For a positive integer n, let

F n ( X , Y ) = k = 1 n ( X sin k π n Y cos k π n ) .

In 2000 Bean and Laugesen proved that for every n3 the area bounded by the curve |Fn(x,y)|=1 is equal to 411nB121n,12, where B(x,y) is the beta function. We provide an elementary proof of this fact based on the polar formula for the area calculation. We also prove that

F n ( X , Y ) = 2 1 n Y 1 k n k  is odd ( 1 ) k 1 2 n k X n k Y k

and demonstrate that n=2n1ν2(n) is the smallest positive integer such that the binary form nFn(X,Y) has integer coefficients. Here ν2(n) denotes the 2-adic order of n.

Citation

Download Citation

Anton Mosunov. "On the area bounded by the curve $\prod_{k = 1}^n\left|x\sin\frac{k\pi}{n} - y\cos\frac{k\pi}{n}\right|=1$." Rocky Mountain J. Math. 50 (5) 1773 - 1777, October 2020. https://doi.org/10.1216/rmj.2020.50.1773

Information

Received: 1 February 2020; Accepted: 9 February 2020; Published: October 2020
First available in Project Euclid: 5 November 2020

zbMATH: 07274833
MathSciNet: MR4170685
Digital Object Identifier: 10.1216/rmj.2020.50.1773

Subjects:
Primary: 11D75, 51M25
Secondary: 11J25, 33B15

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.50 • No. 5 • October 2020
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