Võrdearvud

Võrdearvud, also known in English as congruent numbers, are positive integers n for which there exists a right triangle with rational side lengths and area n. Equivalently, there exists a right-angled triangle with sides a, b, c in the rational numbers such that ab/2 = n and a^2 + b^2 = c^2.

Mathematically, the problem can be reformulated in terms of elliptic curves: n is a võrdearv if and only if the elliptic curve E_n defined by y^2 = x^3 − n^2 x has a nontrivial rational point, which is related to the curve having positive rank over the rational numbers. This connection places võrdearvud at the intersection of elementary geometry and deep areas of number theory, including the theory of elliptic curves and L-functions.

The Congruent Number Problem asks which integers n are võrdearvud. There is no simple classification or formula known; determining membership typically involves computational methods and the exploration of the arithmetic of E_n. Results rely on advanced tools such as descent techniques, modular forms, and the Birch and Swinnerton-Dyer conjecture. Ongoing research combines theory and computation to identify new congruent numbers and to study the distribution of võrdearvud among the positive integers.

See also: Pythagorean triples, elliptic curves, Diophantine equations, modular forms, Birch and Swinnerton-Dyer conjecture.