The Sciences, March/April, 1996: "Beyond the Last Theorem." On Diophantine equations, by mathematician Dorian Goldfeld. To mathematicians, the statement and proof of the STW conjecture were as revolutionary as the first mingling of waters in the Panama Canal. Until that point, the mathematics of elliptic functions and the mathematics of rigid motions had developed in isolation from each other and in strikingly different ways. The study of elliptic curves was a branch of number theory, small, specialized and provincial -- not unlike the study of Diophantine equations. In contrast, the study of rigid motions was a bustling, sophisticated suburb of topology, geometry and analysis, with many applications to engineering and physics. Mathematicians had been working on rigid motions intensely for a hundred years and had accumulated a vast armamentarium of powerful mathematical machinery. By suggesting that the two fields could be linked, Shimura, Taniyama and Weil delivered that heavy machinery to the construction site of elliptic curves; by proving that the link held, Wiles and Taylor started the engines. The result has been a frenzy of productive mathematical work that has benefited each field and is likely to lead to solutions of outstanding problems in other fields as well. ... If the ABC conjecture yields, mathematicians will find themselves staring into a cornucopia of solutions to long-standing problems. Some of those problems are of more than theoretical interest. Nowadays many methods of ensuring the security of electronic mail and other computerized transactions depend heavily on number theory, as programmers develop ciphers based on time-consuming problems in arithmetic. For example, a highly popular technique depends on the difficulty of determining all the large prime factors of a very large number. In principle, it should also be straightforward to create a cipher based on the difficulty of solving problems in Diophantine analysis. The major hurdle is the solvability barrier: the number of variables above which a Diophantine equation becomes impervious to attack. Any cipher based on an equation with that many variables should be absolutely secure. But where is the threshold? All anyone knows is that it probably lies between three and nine variables. At current or foreseeable processing speeds, a nine-variable cipher is impracticably slow, even for the fastest computers. A four-variable Diophantine cipher, however, would be both practical and extremely useful. DIO_fan (35 kb)