The ASU mathematician provides a second example. Prime means that 67 cannot be written as the product of two smaller numbers. The difficulty of factoring is the basis for an important technique called public key encryption, a method for secure data transmission. The fundamental difficulty in both problems is that the number of possibilities is extraordinarily large. The complications that come with size are not artificial.
Large problems occur quite naturally in the real world.
Optimization problems with more than 1,000 variables occur routinely in economic modeling, marketing, weather forecasting, telecommunication networks, microchip design, airline operations, manufacturing, and city planning. To understand the critical role played by the size of a problem, Trotter suggests considering the challenge of determining whether there is a tour visiting all vertices in a graph exactly once. If all possible pairs of nodes are linked by an edge, then the answer is yes. But only certain pairs are linked. One possible approach is to use a high speed computer to list all possible sequences of vertices, then test them one by one to see if each consecutive pair of nodes is linked by an edge.