Not a third number theory, but a third volume, the final in a series on number theory that applies fundamental methods of modern number theory to elucidate the mystery of prime numbers. It looks at modular forms and Iwasawa theory, and illustrates them with many examples for non-specialists and beginners. The introduction reviews developments from before Fermat to his famous last theorem. Other topics are rational points on elliptic curves, conics and the p-adic numbers, the zeta function, and algebraic number theory. Exercises and questions are provided, along with answers. The Japanese original was published by Iwanami Shoten Publishers, Tokyo, in 1998 and in a second edition in 2005. The translation is by Masato Kuwata. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com)