We formally define polynomial endofunctors on the category of sets, referring to them as polynomial functors or simply polynomials. These are constructed as sums of representable functors on the category of sets. We provide concrete examples of polynomials and highlight that the set of representable summands of a polynomial is isomorphic to the set obtained by evaluating the functor at the singleton set, which we term the positions of the polynomial. For each position, the elements of the representing set of the corresponding representable summand are called the directions. Beyond representables, we define three additional special classes of polynomials: constants, linear polynomials, and monomials. We close the chapter by offering three intuitive interpretations of positions and directions: as menus and options available to a decision-making agent, as roots and leaves of specific directed graphs called corolla forests, and as entries in two-cell spreadsheets we refer to as polyboxes.

We show that the category of comonoids, defined with respect to the composition product in the category of polynomial functors, is equivalent to a category of small categories as objects but with an interesting type of morphism called retrofunctors. Unlike traditional functors, retrofunctors operate in a forward-backward manner, offering a different kind of relationship between categories. We introduce this concept of retrofunctors, provide examples to illustrate their behavior, and explain their role in modeling state systems.

We model discrete-time dynamical systems using a specific class of lenses between polynomials whose domains are equipped with a bijection between their positions and their directions. We introduce Moore machines and deterministic state automata as key examples, showing how these morphisms describe state transitions and interactions. We also explain how to build new dynamical systems from existing ones using operations like products, parallel composition, and compositions of these maps. This chapter demonstrates how polynomial functors can be used to represent and analyze discrete-time dynamical behavior in a clear, structured way.

We describe a range of additional category-theoretic structures on the category of polynomial functors. These include concepts like adjunctions, epi-mono factorizations, and cartesian closure. We also cover limits and colimits of polynomials and explore vertical-cartesian factorizations of morphisms. The chapter highlights how these structures provide new tools for working with polynomial functors and extend their usefulness in modeling various types of interactions and constructions.

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.