Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-18T04:13:46.900Z Has data issue: false hasContentIssue false
  • English
  • Français

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 [email protected]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bácskái, Z. F., Buchholz, R. H., Rathbun, R. L. and Smith, M. J., Symmetries of triangles with two rational medians, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.6533, (2003).Google Scholar
Buchholz, R. H., ‘On triangles with rational altitudes, angle bisectors and medians’, PhD Thesis, University of Newcastle, Australia, 1989.Google Scholar
Buchholz, R. H. and Rathbun, R. L., ‘An infinite set of heron triangles with two rational medians’, Amer. Math. Monthly 104(2) (1997), 106–115.10.1080/00029890.1997.11990608Google Scholar
Buchholz, R. H. and Rathbun, R. L., ‘Heron triangles and elliptic curves’, Bull. Aust. Math. Soc. 58 (1998), 411–421.10.1017/S0004972700032391Google Scholar
Buchholz, R. H., ‘Triangles with three rational medians’, J. Number Theory 97(1) (2002), 113–131.10.1006/jnth.2002.2777Google Scholar
Buchholz, R. H. and Stingley, R. P., Heron triangles with three rational medians, https://ca827bd0-a-62cb3a1a-s-sites.googlegroups.com/site/teufelpi/papers/D21.pdf?attachauth=ANoY7cpsOhPDAmImchFC8BEi, (2013).Google Scholar
Cohen, H., Number Theory: Volume II: Analytic and Modern Tools, 2nd edn (Springer, New York, 2007).Google Scholar
Dickson, L. E., History of the Theory of Numbers (Carnegie Institution, Washington, 1919).Google Scholar
Dujella, A. and Peral, J. C., ‘Elliptic curves and triangles with three rational medians’, J. Number Theory 133 (2013), 2083–2091.10.1016/j.jnt.2012.12.003Google Scholar
Dujella, A. and Peral, J. C., ‘Elliptic curves coming from heron triangles’, Rocky Mountain J. Math. 44(4) (2014), 1145–1160.10.1216/RMJ-2014-44-4-1145Google Scholar
Euler, L., ‘Solutio facilior problematis Diophantei circa triangulum in quo rectae ex angulis latera opposita bisecantes rationaliter exprimantur’, Mem. Acad. Sci. St. Petersburg 2 (1810), 10–16; [See L. Euler, Opera Omnia, Commentationes Arithmeticae 3, paper 732 (1911)].Google Scholar
Faltings, G., ‘Endlichkeitsästze für abelsche Varietten ber Zahlkörpern’, Invent. Math. 73(3) (1983), 349–366.10.1007/BF01388432Google Scholar
Guy, R., Unsolved Problems in Number Theory (Springer, New York, 1981).10.1007/978-1-4757-1738-9Google Scholar
Rathbun, R., Perfect rational triangle, https://www.ics.uci.edu/∼eppstein/junkyard/q-triangle.html, (1987).Google Scholar
Silverman, J. H., Graduate Texts in Mathematics: The Arithmetic of Elliptic Curves, 2nd edn (Springer, New York, 2009).10.1007/978-0-387-09494-6Google Scholar
Stanica, P., Sarkar, S., Gupta, S. S., Maitra, S. and Kar, N., ‘Counting Heron triangles with constraints’, J. Integers 13 (2013), A3.Google Scholar