1. On certain Diophantine equations concerning the area of right triangles.
- Author
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Zhang, Yong and Gao, Dan
- Subjects
- *
TRIANGLES , *ELLIPTIC curves , *SUM of squares , *INFINITY (Mathematics) , *DIOPHANTINE equations - 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. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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