On certain Diophantine equations concerning the area of right triangles
-
Yong Zhang
and
Dan Gao
Abstract
Using the theory of elliptic curve, we show that all right triangles, such that the sum of the area and the square of the sum of legs is a square, are given by an infinite set. Similarly, we get all right triangles such that the sum of the area and the square of the semi-perimeter is a square. Using the theory of Pell’s equation, we prove that there are infinitely many non-primitive right triangles such that the sum of the area and the hypotenuse (or the smaller leg) is a square, and an infinity of primitive right triangles such that the sum of the area and the smaller leg (or the perimeter, the semi-perimeter, the larger leg) is a square.
(Communicated by István Gaál)
Acknowledgement
The authors would like to thank the anonymous referee for giving valuable comments and suggestions.
References
[1] Cohen, H.: Number Theory, Vol. I: Tools and Diophantine Equations. Grad. Texts in Math. 239, Springer, New York, 2007.Search in Google Scholar
[2] Dickson, L. E.: History of the Theory of Numbers, Vol. II: Diophantine Analysis Dover Publications, New York, 2005.Search in Google Scholar
[3] Heath, T. L.: Diophantus of Alexandria: A Study in the History of Greek Algebra Cambridge University Press, Cambridge, 1910.Search in Google Scholar
[4] Koshy, T.: Fibonacci and Lucas Numbers with Applications Wiley, New York, 2001.10.1002/9781118033067Search in Google Scholar
[5] Macleod, A. J.: Two extreme Diophantine problems concerning the perimeter of Pythagorean triangles Glas. Mat. Ser. III 46(1) (2011), 1–5.10.3336/gm.46.1.01Search in Google Scholar
[6] Mordell, L. J.: Diophantine Equations. Pure Appl. Math., Vol. 30, Academic Press, London, 1969.Search in Google Scholar
[7] Skolem, T.: Diophantische Gleichungen Chelsea, 1950.Search in Google Scholar
© 2021 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular papers
- Prof. RNDr. Ing. Lubomír Kubáček, DrSc., Dr.h.c. –Nonagenarian
- Doc. RNDr. Roman Frič, DrSc. passed away
- Outer and inner approximations in quantum spaces
- Linear derivations on Banach *-algebras
- New fractional order discrete Grüss type inequality
- Exponential trigonometric convex functions and Hermite-Hadamard type inequalities
- Generalized Minkowski type inequality for pseudo-integral
- Study of the Q-spiral-like functions of complex order
- Radius of starlikeness of certain analytic functions
- Successive approximations for a differential equation in a Banach space via Constantin condition
- Approximation of the multi-m-Jensen-quadratic mappings and a fixed point approach
- Oscillation and asymptotic behavior of a higher-order neutral delay difference equation with variable delays under Δm
- Sequence selection properties in Cp(X) with the double ideals
- On the paranormed Nörlund difference sequence space of fractional order and geometric properties
- On certain Diophantine equations concerning the area of right triangles
- Weighted projective Ricci curvature in Finsler geometry
- Euler classes of vector bundles over manifolds
- A new generalized Lindley-Weibull class of distributions: Theory, properties and applications
- Dynamical behaviors of a prey-predator model with foraging arena scheme in polluted environments
- The (α, β)-ramification invariants of a number field
Articles in the same Issue
- Regular papers
- Prof. RNDr. Ing. Lubomír Kubáček, DrSc., Dr.h.c. –Nonagenarian
- Doc. RNDr. Roman Frič, DrSc. passed away
- Outer and inner approximations in quantum spaces
- Linear derivations on Banach *-algebras
- New fractional order discrete Grüss type inequality
- Exponential trigonometric convex functions and Hermite-Hadamard type inequalities
- Generalized Minkowski type inequality for pseudo-integral
- Study of the Q-spiral-like functions of complex order
- Radius of starlikeness of certain analytic functions
- Successive approximations for a differential equation in a Banach space via Constantin condition
- Approximation of the multi-m-Jensen-quadratic mappings and a fixed point approach
- Oscillation and asymptotic behavior of a higher-order neutral delay difference equation with variable delays under Δm
- Sequence selection properties in Cp(X) with the double ideals
- On the paranormed Nörlund difference sequence space of fractional order and geometric properties
- On certain Diophantine equations concerning the area of right triangles
- Weighted projective Ricci curvature in Finsler geometry
- Euler classes of vector bundles over manifolds
- A new generalized Lindley-Weibull class of distributions: Theory, properties and applications
- Dynamical behaviors of a prey-predator model with foraging arena scheme in polluted environments
- The (α, β)-ramification invariants of a number field
