Right Triangles with Algebraic Sides and Elliptic Curves over Number Fields

Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field Q(λ) (depending on n) and an explicit point Pλ of infinite order in the Mordell-Weil group of the elliptic curve Y 2 = X − nX over Q(λ). 1. Congruent numbers over the rationals A positive integer n is called a congruent number if there exists a right triangle with rational sides and area equal to n, i.e., there exist a, b, c ∈ Q∗ with a + b = c and 1 2 ab = n. (1) It is easy to decide whether there is a right triangle of given area and integral sides (thanks to Euclid’s characterization of the Pythagorean triples). The case of rational sides, known as the congruent number problem, is not completely understood. Fibonacci showed that 5 is a congruent number (since one may take a = 3 2 , b = 20 3 and c = 41 6 ). Fermat found that 1, 2 and 3 are not congruent numbers. Hence, there is no perfect square amongst the congruent numbers since otherwise the corresponding rational triangle would be similar to one with area equal to 1. By the same reason one can restrict to square-free positive integers, a condition we will assume in the sequel. There is a fruitful translation of the congruent number problem into the language of elliptic curves. Suppose n is a congruent number. It follows from (1) that there exist three squares in arithmetic progression of distance n, namely x−n, x, x+n, where x = c/4. Therefore their product (x−n)x(x+ n) is again a rational square. In other words, a right triangle of area n and Date: March 26, 2009. 2000 Mathematics Subject Classification. Primary 11G05; Secondary 11A99. 1