The theory of binary quadratic forms is one of the most cherished mathematical flowers. We shall give its account without any compromise but plainly as much as possible, so that readers will, in particular, acquire the real ability to solve the problem of representing given integers by a given quadratic form. Historically, the theory of quadratic forms prepared the theory of quadratic number fields and beyond through the composition and genus theory. In this chapter, such a viewpoint is carefully attended while using only integers and matrix modules, i.e., without entering into algebraic number theory. This chapter offers an indispensable experience and materials before moving to higher modern algebraic number theory as well as into advanced analytic number theory.

The Fourier analysis is applied to the deviations from divisibility, so that readers experience the first instances of mathematical dualities. Through this duality, we are able to see the congruence issues much better than solely dealing with them naively. The theory of quadratic binary equations yields the quadratic reciprocity law. Gauss’ eight proofs, save for the fifth, of it, as well as Legendre’s, are fully explained from various angles, which, as a whole, indicates the modern algebraic number theory and signifies the theory of primes in arithmetic progressions. We shall also be led inevitably to the history of the theory of algebraic equations, which will be presented with some concrete examples.

This chapter is essentially a treatise on the subject. We shall give all basic material on today’s theory of the distribution of prime numbers; note that we restrict ourselves to asymptotic study, as it is more basic. Thus all deep prime number theorem (Vinogradov’s, Huxley’s, Bombieri’s, Linnik’s) will be presented with complete proofs; analytic means as well are developed with full details. The sieve mechanism stemmed from Linnik and Selberg will be seen as the main mover of our discussion. In the final section, we shall deal with a recent astounding discovery on bounded gaps between prime numbers; our presentation is introductory but will suffice for readers to proceed to the full details.

Congruence theory is the same as the study of the deviations from divisibility. With this view, here a basis of modern number theory is constructed. It is built, however, on the availability of prime power decomposition of moduli, save for the linear case. We shall show this aspect can be reverted to an extent so that the integer factorization, one of the fundamental issues in whole mathematics and also in our digital life, can be discussed with certain success. Our approach also yields a description of a quantum factorization algorithm solely with plain mathematical terms. Further, we shall hint at another great issue, the general reciprocity among power residues, from which the modern algebraic number theory stemmed.

Classical multiplicative number theory with Euclid’s algorithm and continued fractions is presented anew in matrix formulation, which shows immediately, for instance, that there exist group structures over the integers. Very basics of modern sieve methods and prime number theory are also described so that readers can foresee well what will be developed in the analysis oriented final chapter. Continued fractions are presented as a device still fundamental in practical approaches to number theory, despite they are ignored in most modern treatises, which are often written with solely theoretical views. This chapter also describes a great historical tradition or cultural interactions encircling Euclid’s Elements, and how deeply we owe to the efforts of people in past ages.

This volume is an outcome of

The 39th Taniguchi International Symposium on Mathematics

organized by myself, and of its forum, May 20–24, organized by N. Hirata-Kohno, L. Murata and myself as a conference at the Research Institute for Mathematical Sciences, Kyoto University.

I am deeply indebted to the Taniguchi Foundation for the generous support that made the symposium and the conference possible. The organizers of the conference acknowledge sincerely that the speakers were supported in part by the Inoue Science Foundation, the Kurata Foundation, Saneyoshi Scholarship Foundation, the Sumitomo Foundation; College of Science and Technology of Nihon University, the Research Institute for Mathematical Sciences of Kyoto University; and the Grant-Aid for General Scientific Research from the Ministry of Education, Science and Culture (through the courtesy of Prof. M. Koike, Kyushu University).

My special thanks are due to Profs. Hirata-Kohno and Murata for their unfailing collaboration during the three years of difficult preparation for the meetings.