This Small Grant for Exploratory Research supports a project to implement the Number Field Sieve, the newest method for factoring large integers. The principal investigators are Professor Robert Robson of the Mathematics Department and Professor Walter Rudd of the Computer Science Department at Oregon State University, and Professor Joe Buhler of Reed College. The Number Field Sieve has a predicted running time that is so much better than that of the previously known methods of factoring that the Principal Investigators believe that they will be able to factor numbers in the 145-digit range, far larger than has been possible up to now. Except for counting, number theory, which is the study of the properties of the whole numbers, is the oldest branch of mathematics. In modern days, problems in number theory have furnished the driving force to creation of new mathematics in the fields of pure algebra, analysis, and geometry; some of the most recent and most astonishing applications of number theory have appeared in theoretical computer science and coding theory. In particular, the difficulty or ease of factoring large numbers has turned out to have a direct application to the security of public-key codes.