Solutions of Indefinite Equations

Indefinite equation is an unsolved problem in number theory. Through ex-ploration, the author has been able to use a simple elementary algebraic method to solve the solutions of all three variable indefinite equations. In this paper, we will introduce and prove the solutions of Pythagorean equation, Fermat’s theorem, Bill equation and so on.


Definition of Indefinite Equation
We also know indefinite equation (also called Diophantine equation) that all unknowns and known numbers are positive integers [1] [2]. The indefinite equation, especially the higher-order indefinite equation, is a difficult problem which has not been solved thoroughly in number theory. In this paper, we will introduce their new solutions one by one. Theorem 1. Here is the three variable indefinite equation as

First Order Indefinite Equation Some Theorems
If one of the terms is an arbitrary positive integer, the Equation (1) must have a solution.
Proof. Suppose C is any positive integer, A and B are two positive integers, then the sum of A and B must be positive integers. We can use the number axis to verify that C is also a positive integer. □ Advances in Pure Mathematics proved by many kinds of proofs [3]. Namely Pythagorean theorem is However, its simple way of solutions has never been seen. Next we will find and prove the solutions.
Theorem 2. If positive integers a, b, c are a series of positive integer solutions of following equation

Definition of higher degree indefinite equation is an indefinite equation in
which the exponents of all powers are greater than or equal to 3, as n n n where n ≥ 3, x ≥ 3, y ≥ 3, z ≥ 3.

Fermat's Last Theorem
We have known that Fermat's last theorem has been proved by British Mathematician Andew. Wiles using the properties of elliptic curves (1993) [5]. Now we present a proof using elementary algebra as following.
Theorem 3 (Fermat's Last Theorem). If A, B, C are tree positive integers, n ≥ 3, than equation is no integer solutions. Proof. Suppose Equation (7) By Equations (7) and (10) Comparisons Equations (11) and (12), we have: (12) is true, also need q = C, and we obtain that if Equation (12) is true, require the following are true: Obviously, these equations are contradictory by n ≥ 3, it is impossible. So, the hypothesis is not valid, the Equation (11) is not true, and the Equation (7)   . □ Therefore, the equation as (7) is no integer solutions.

Beal's Conjecture
Beal's conjecture: if the indefinite equation is true, there , , x y y z x z ≠ ≠ ≠ and 3, 3, 3 x y z ≥ ≥ ≥ ; than A, B, C must have a common factor. We have proved that Beal's conjecture is true in the paper of "Proof of Beal Conjecture" [6].

Solutions by L-Algorithm
The equations as Equation (14) (14). This is solution for the Beal equations. Therefore, we can use the above method to solve other indefinite equations. Example 1. Solve the following indefinite equation A B C + =. (16) Firstly, a, b, d are selected to satisfy a x + b y = c, as

Conclusion
Through the above introduction, we understand the new solution of the indefinite equation, which shows that the indefinite equation can be solved by the method of elementary algebra. According to this method flexibly, we will solve more higher degree indefinite equations. It adds a new way to solve the higher order indefinite equation for number theory.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.