A positive integer is called a congruent number if it is the area of some right triangle with rational side-lengths. For example,  5, 6, 7 are congruent numbers since they are area of right triangles with side lengths (3/2, 20/3, 41/6), (3, 4, 5), and (35/12, 24/5, 337/60), respectively. Fermat showed that 1, 2, 3 are not congruent by his famous infinite descent method.  The congruent number problem is to determine whether a given integer is a congruent number.  It has a thousand years of history and has closed relation to the Birch and Swinnerton-Dyer (BSD, for short) conjecture. The BSD conjecture is one of seven millennium problems listed by the Clay Mathematical Institute. It predicts a deep relation between the L-function of an elliptic curve over a number field and its Mordell-Weil group.  The BSD conjecture predicts that any positive integer, congruent to 5, 6, or 7 modulo 8, is a congruent number. In this talk, we will introduce some recent progress on the BSD conjecture and the congruent number problem.

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