Department of Mathematics
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A positive integer n is called a congruent number if there is a rational right triangle with area n. The 2000-year old congruent number problem asks for a complete description of all congruent numbers—it is still open. Of course, using the well-known description of the Pythagorean triples, it is certainly possible to write a dull algorithm to mechanically produce a list of congruent numbers - but there is no guarantee that a number is not a congruent number if it does not show up in the list because it might eventually turn up if we wait long enough. In this talk we show a fundamental connection between the congruent number problem and the rational points on certain cubic curves which ultimately leads to a solution of the congruent number problem (modulo, of course, a famous and a deeper conjecture of Birch and Swinnerton-Dyer).

References:

1. N. I. Koblitz, Introduction to elliptic curves and modular forms, Second edition, Springer, New York, 1993
2. Coates, John H. Congruent number problem. Q. J. Pure Appl. Math. 1 (2005), no. 1, 14–27.
3. Top, Jaap; Yui, Noriko;. Congruent number problems and their variants. Algorithmic number theory: lattices, number fields, curves and cryptography, 613–639, Math. Sci. Res. Inst. Publ., 44, Cambridge Univ. Press, Cambridge, 2008.