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PERFECT TRIANGLES ON THE CURVE $C_{4}$

Part of: Computational number theory

Published online by Cambridge University Press:  09 October 2019

SHAHRINA ISMAIL Open the ORCID record for SHAHRINA ISMAIL [Opens in a new window]
SHAHRINA ISMAIL*
Affiliation:
School of Mathematics and Physics, The University of Queensland, St Lucia, Qld 4072, Australia email s.ismail1@uq.edu.au
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Abstract

A Heron triangle is a triangle that has three rational sides $(a,b,c)$ and a rational area, whereas a perfect triangle is a Heron triangle that has three rational medians $(k,l,m)$. Finding a perfect triangle was stated as an open problem by Richard Guy [Unsolved Problems in Number Theory (Springer, New York, 1981)]. Heron triangles with two rational medians are parametrized by the eight curves $C_{1},\ldots ,C_{8}$ mentioned in Buchholz and Rathbun [‘An infinite set of heron triangles with two rational medians’, Amer. Math. Monthly 104(2) (1997), 106–115; ‘Heron triangles and elliptic curves’, Bull. Aust. Math.Soc. 58 (1998), 411–421] and Bácskái et al. [Symmetries of triangles with two rational medians, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.6533, 2003]. In this paper, we reveal results on the curve $C_{4}$ which has the property of satisfying conditions such that six of seven parameters given by three sides, two medians and area are rational. Our aim is to perform an extensive search to prove the nonexistence of a perfect triangle arising from this curve.

Keywords

elliptic curveperfect trianglep-adic

MSC classification

Primary: 11Y05: Factorization
Secondary: 11Y35: Analytic computations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 1 , August 2020 , pp. 68 - 80
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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References

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