1, introduction of indefinite equation.
Indefinite equation refers to an equation or system of equations whose solution range is integer, positive integer, rational number or algebraic integer, and the number of unknowns is usually more than the number of equations.
Diophantine, an ancient Greek mathematician, studied several such equations at the beginning of the third century, so the indefinite equation is also called Diophantine equation, which is an important branch of number theory and one of the most active mathematical fields in history. The content of indefinite equation is very rich, which is closely related to algebraic number theory, geometric number theory and set theory.
2. Study the problem of indefinite equation.
There are three problems to be solved in learning indefinite equations: judging when there is a solution; Determine the number of solutions when there are solutions; Find all the solutions.
China was the first country to study indefinite equations, and the problem of "Five Well" at the beginning of A.D. was an indefinite equations problem. In the 5th century, China made a systematic study on the theory of indefinite equations. Qin's "great solution" links indefinite equations with congruence theory.
Types of indefinite equations:
1, binary linear indefinite equation.
The general form of binary linear indefinite equation is ax+by = C, where a, b and c are integers and ab≠0. The necessary and sufficient condition for this equation to have an integer solution is that the greatest common divisor of A and B can be divisible by C. Regarding the integer multivariate linear indefinite equation, there can be related methods such as matrix solution and program design to help solve it.
2. Binary quadratic indefinite equation.
In essence, the binary quadratic indefinite equation can be summed up as the problem of finding the rational point or integral point of quadratic curve (that is, conic curve).
3. High-order indefinite equation.
For indefinite equations higher than quadratic, it is quite complicated. When n >; At 2 o'clock, x n+y n = z n has no nontrivial integer solution, which is the famous Fermat's last theorem, which has been proved by the British mathematician andrew willis for three centuries.
Multivariate high-order indefinite equations have no general solution, and any solution can only solve some special indefinite equations. The integer solutions of some special indefinite equations are discussed by using quadratic field. Commonly used solutions include algebraic identity deformation, inequality estimation, congruence, construction and infinite recursion.