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A Heron triangle and a Diophantine equation

Abstract

Given any positive integer n, it is well known that there always exists a triangle with rational sides ab and c such that the area of the triangle is n. For any pair of primes (pq) such that \(p \not \equiv 1\) (mod 8) and \(p^{2}+1=2q\), we look into the possibility of the existence of triangles having rational sides with p as the area and \(p^{-1}\) as \(\tan \frac{\theta }{2}\) for one of the angles \(\theta \). We also discuss the relation of such triangles with the solutions of certain Diophantine equations.

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Acknowledgements

The authors would like to thank Prof. Anupam Saikia for providing valuable suggestions throughout this work. The authors are also thankful for the valuable suggestions made by the anonymous referee. The first author would like to acknowledge the fellowship and amenities provided by the Council of Scientific and Industrial Research, India (CSIR) and BITS-Pilani, Hyderabad. The second author would like to thank IIT-Guwahati for providing the fellowship and amenities. The third author was supported by BITS-Pilani, Hyderabad.

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Correspondence to Debopam Chakraborty.

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Ghale, V., Das, S. & Chakraborty, D. A Heron triangle and a Diophantine equation. Period Math Hung (2022). https://doi.org/10.1007/s10998-022-00491-5

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  • DOI: https://doi.org/10.1007/s10998-022-00491-5

Keywords

  • Elliptic curve
  • Heron triangle
  • Diophantine equation

Mathematics Subject Classification

  • Primary 11G05
  • 51M04
  • Secondary 11D25