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such as the four-color problem first proposed in the mid-19th century. The theorem stated that four colors are sufficient to color any map, given that any two countries with a contiguous boundary require different colors. The theorem was finally proved in 1976 by means of a large-scale computer at the University of Illinois.

Mathematical knowledge in the modern world is advancing at a faster rate than ever before. Theories that were once separate have been incorporated into theories that are both more comprehensive and more abstract. Although many important problems have been solved, other hardy perennials, such as the Riemann hypothesis, remain, and new and equally challenging problems arise. Even the most abstract mathematics seems to be finding applications.



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