1. divisibility theory. This paper introduces the basic concepts such as divisibility, factor, multiple, prime number and composite number. The main achievements of this theory are: unique decomposition theorem, Pei tree theorem, Euclid's transition division, fundamental theorem of arithmetic, proof of infinite prime number, etc.
2. Congruence theory. Mainly from Gaussian arithmetic research. The concepts of congruence, primitive root, exponent, square residue and congruence equation are defined. Main achievements: Quadratic Reciprocity Law, euler theorem, Fermat's Last Theorem, Wilson's Theorem, Sun Tzu's Theorem (China's Remainder Theorem), etc.
3. Continued fraction theory. The concept and algorithm of continued fraction are introduced. In particular, the continued fraction expansion of integer square root is studied. Main achievements: cyclic continued fraction expansion, optimal approximation problem, solving Pell equation.
4. Indefinite equation. This paper mainly studies the indefinite equations corresponding to the lower algebraic curve, such as the quotient height theorem of Pythagorean equation and the continued fraction solution of Pell equation. It also includes the solution of the fourth Fermat equation and so on.
5. Number theory function. Such as Euler function, Mobius transformation and so on.
6. Gaussian function. The first level is called mathematical concept, which is a form of thinking that reflects the essential attributes of objects. In the process of cognition, human beings rise from perceptual knowledge to rational knowledge, and abstract and summarize the essential characteristics of perceptual things, which becomes a concept. The language form of expressing concepts is words or phrases. Scientific concepts, especially mathematical concepts, are more stringent, and at least three conditions must be met: specificity, accuracy and verifiability. For example, "twin prime numbers" is a mathematical concept.
The second level is called mathematical proposition, which is a sentence to judge the relationship between a series of mathematical concepts. A proposition is either true or false (this is guaranteed by law of excluded middle in logic). A true proposition contains theorems, lemmas, inferences, facts, etc. A proposition can be an existential proposition (expressed as "being …") or a full-name proposition (expressed as "for everything …"). The third level is called mathematical theory, which combines methods, formulas, axioms, theorems and principles into a system. For example, "elementary number theory" consists of axioms (such as equality axioms), theorems (such as Fermat's last theorem), principles (such as one-to-one correspondence principle of pigeon hole principle) and formulas. In mathematical proof, the full name proposition can not always be judged by enumeration, because mathematics sometimes faces infinite objects, and it can never enumerate every situation one by one. Incomplete induction is not feasible in mathematics, which only recognizes deductive logic (mathematical induction, transfinite induction, etc.) ).