In fact, we can show these three points are independent, and that the rank of E t over ℚ( t) is ≥ 3 for all but finitely many values of t. Using the theory of elliptic curves we are able to prove that there are infinitely many examples of each type. We then consider Heron triangle and integer rhombus pairs. The first problem we examine regards integer isosceles triangles and integer parallelograms which share a common area and common perimeter. In this paper we continue this line of study. Zhang, and integer right triangle and rhombus pairs by S. Guy and Bremner, integer right triangle and parallelogram pairs by Y. Bremner, Heron triangle and rectangle pairs by R.
Several other works in this direction have been solved, all involving pairs of geometric shapes having a common area and common perimeter: two distinct Heron triangles by A. In that same paper, Guy also showed that there are infinitely many such isosceles triangle and rectangle pairs. In 1995, Guy showed that the answer was affirmative, but that there is no non-degenerate right triangle and rectangle pair with the same property. Guy if there were triangles with integer sides associated with rectangles having the same perimeter and area. Īnother problem connecting geometrical objects with number theory is devoted to the construction of triangles with area, perimeter or side lengths with certain arithmetic properties. similarly studied curves arising from Brahmagupta quadrilaterals. In Naskrecki constructed elliptic curves associated to Pythagorian triplets, and Izadi et. Izadi, Khoshnam, and Moody later generalized their notions to Heron quadrilaterals. Both Goins and Maddox and Dujella and Peral constructed elliptic curves over ℚ coming from Heron triangles. For example, there is the well-known congruent number problem which asks: given a positive integer n, does there exist a right triangle with rational side lengths whose area is n? As a second example, several researchers have related various types of triangles and quadrilaterals to the theory of elliptic curves. There are many questions in number theory which are related to triangles, rectangles, squares, polygons, and so forth.
The study of geometrical objects is a very ancient problem.
0 kommentar(er)
