Theorem 1. The equation ax+by = c has a solution if and only if (a, b) | c;
Theorem 2. If (a, b) = 1 and x_0 and y_0 are a solution of ax+by = c, all solutions of the equation can be expressed as
Theorem 3. Necessary and Sufficient Conditions for the N-dimensional linear indefinite equation A _ 1 x _1+A _ 2x _ 2+…+A _ NX _ n = c, (A _1,A _ 2, …a_n, c∈N) to have a solution.
Methods and skills:
1. To solve a binary linear indefinite equation, it is usually necessary to determine whether the equation has a solution. If there is a solution, you can first find the special solution of ax+by = c, and then write the general solution. When the coefficient of the indefinite equation is not large, sometimes the solution can be obtained by observation, that is, introducing variables and gradually reducing the coefficient until it is easy to get its special solution;
2. When solving the linear indefinite equation A _ 1 X _1+A _ 2x _ 2+…+A _ NX _ n = C, we can solve (A _1,a_2) = d_2, (d_2, A _ 2) in turn. If c is divisible by d_n, the equation has a solution, and it is a set of equations:
Find all the solutions of the last equation, then substitute every value of t_(n- 1) into the penultimate equation, find all its solutions, and so on, all the solutions of the equation can be obtained.
3. A system of equations consisting of m linear indefinite equations with n variables, where m
2. congruence method: if the indefinite equation F( x_ 1, x_2, …, x_n) = 0 has an integer solution, then for any m∈N, its integer solution (x_ 1, x_2, …, x_n) satisfies f (.
3. Inequality estimation method: using inequality tools to determine the range of some letters in the indefinite equation, and then solving them separately;
4. Infinite descent method: If the proposition P(n) about positive integer n holds for some positive integers, let n_0 be the smallest positive integer that makes P(n) hold, it can be deduced that there is a positive integer n, so that n _ 1
Methods and skills:
1. Factorization is the most basic method in indefinite equations, and its theoretical basis is the unique decomposition theorem of integers. As a means of solving problems, factorization has no definite procedure to follow, and only through concrete examples can we have a profound understanding.
2. The congruence method is mainly used to prove that the equation has no solution or the necessary conditions for deriving a solution, so as to prepare for further solution or verification. The key to congruence is to choose the right module, which needs many attempts;
3. Inequality estimation method is mainly aimed at an equation that has an integer solution, so there must be a real solution. When the real solution of the equation is a bounded set, it is important to consider that there are at most a limited number of integer solutions in a limited range, and check them one by one to find all the solutions. If the real number solution of the equation is unbounded, it is necessary to take the integer as the center and use various properties of the integer to generate applicable inequalities.
4. The core of the argument of infinite descent method is to try to construct a new solution of the equation, so that it is "strictly less than" the selected solution, which leads to contradictions. 1. The basic idea of finding integer solution of indefinite equation ax+by = cxy (abc≠0) by decomposition method;
Ax+by = cxy is transformed into (x-a)(cy -b) = ab, and if ab can be decomposed into ab = a _1b _1= a _ 2b _ 2 = … = a _ ib _ i ∈ z, the general form of the solution is.
2. Definition 2: The equation of x 2+y 2 = z 2 is called Pythagorean number equation, where x, y and z are positive integers.
For the equation x 2+y 2 = z 2, if (x, y) = d, then d 2 | z 2, so we only need to discuss the case of (x, y) = 1. At this time, it is easy to know that x, y and z are paired, and this paired positive integer group is called the primitive solution of the equation.
Theorem 3. All solutions of Pythagorean equation satisfying condition 2|y can be expressed as:
Where a > b>0, (a, b) = 1, and a and b are even and odd numbers.
Inference: All positive integer solutions of Pythagorean equation (the order of X and Y is indistinguishable) can be expressed as:
| where a > b>0 is a pair of positive integers with different parity, and d is an integer.
The integer solution of Pythagorean indefinite equation is mainly solved according to theorem.
3. Definition 3. Equation x 2-dy 2 =1,4 (x, y∈Z, positive integer d is not a square number) is a special case of x 2-dy 2 = c, which is called Pell equation.
This kind of bivariate quadratic equation is complicated, which essentially boils down to the study of hyperbolic equation x 2-dy 2 = c, where c and d are integers, d >；; 0 is not a square number, c ≠ 0. Mainly used to prove that the problem has countless integer solutions. For a specific d, a positive integer solution can be obtained by trial and error. If the above Pell equation has a positive integer solution (x, y), it is said that the minimum positive integer solution of x+yd 0.5 is its minimum solution.
Theorem 4. Pell equation x x 2-dy 2 =1(x, y∈Z, positive integer d is not a square number) must have a positive integer solution. If its minimum solution is (x_ 1, y_ 1), all its solutions can be expressed as:
The above formula can also be written in the following form:
Theorem 5. Pell equation x 2-dy 2 =-1(x, y∈Z, positive integer d is not a square number) has either no positive integer solution or an infinite number of positive integer solutions. In the latter case, let its minimum solution be (x_ 1, y _ 65438+).
| Theorem 6. (Fermat's last theorem) The equation x n+y n = z n (n ≥ 3 n ≥ 3 and an integer) has no positive integer solution.
The proof of Fermat's Last Theorem has always been a difficult problem in mathematics, but in June 1994, A. wiles, a professor of mathematics at Princeton University in the United States, completely solved this problem. At this point, this mathematical problem that has puzzled people for more than 400 years has finally revealed its true face and taken off its mystery.