1. Definition of Indefinite Equation
We also know indefinite equation (also called Diophantine equation) that all unknowns and known numbers are positive integers  . 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.
2. First Order Indefinite Equation Some Theorems
Theorem 1. Here is the three variable indefinite equation as
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. □
3. Pythagorean Equation (Pythagorean Theorem)
Pythagorean theorem is the most familiar mathematical formula that has been proved by many kinds of proofs . Namely Pythagorean theorem is
However, its simple way of solutions has never been seen. Next we will find and prove the solutions.
By Equation (2), we get
By C − B = d, d = 1, 2, 3, ∙∙∙, we will derive many solutions to Equation (2), as Table 1.
Very regular! It is not difficult to sum it up that general solutions are: A = 3k, B = 4k, C = 5k, k = 1, 2, 3, ∙∙∙.
Theorem 2. If positive integers a, b, c are a series of positive integer solutions of following equation
when c − b = 1.
Then as A = ka, B = kb, C = kc also are solutions of equation as A2 + B2 = C2.
Proof. Suppose Equation (3) is tenable. Multiplying k2 on both sides of the Equation (3), we obtain
So the solutions of A2 + B2 = C2 are A = ka, B = kb, C = kc. □
In addition, we also proof when (C − B) = 1, have array solutions as A = 2m + 1, B = 4m(m + 1), C = B + 1.
They are called original numbers of shares . Obviously, (3) if there is a solution of array (c − b) = 1, then (2) also has countless series of positive integer solutions. In this way, we can find the solution of the infinite Pythagorean equations.
4. Higher Degree Indefinite 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
Table 1. A table of A, B, C, C − B.
where n ≥ 3, x ≥ 3, y ≥ 3, z ≥ 3.
4.1. 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) . 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) is true. Let A = C − a, B = C − b, then
(8) + (9):
By Equations (7) and (10), we have
Comparisons Equations (11) and (12), we have:
So, by theorem 3, if Equation (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) also is not true. □
Theorem 3. If the following indefinite equation is holds, need C = q, if n ≥ 3:
Proof. Obviously, when n ≥ 3, only have C = q to get polynomial
Therefore, the equation as (7) is no integer solutions.
4.2. Beal Equation
4.2.1. Beal’s Conjecture
Beal’s conjecture: if the indefinite equation
is true, there and ; 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” .
4.2.2. Solutions by L-Algorithm
The equations as Equation (14) are called Beal equations. Let’s find their solutions by L-algorithm.
By Theorem 1, there is always ax + by = c (it is called original equation of the Equation (14)) than multiplying the least common factor cs on both sides of original equation
Lest s = xy, than
So, if z|(xy + 1), or xy + 1 = kz, then A = acy, B = bcx, C = ck are solutions for the Equation (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
Firstly, a, b, d are selected to satisfy ax + by = c, as
Then, choose cs, let s = xy = 35, according to (15), multiplying 531235 on the two sides of (17):
So, that solutions of Equation (16) are A = 5 × 53127, B = 5 × 53125, C = 531212. □
Example 2. Solve the following indefinite equation
By Example 1, we know the original of Equation (18) is
Thus, by Equation (20), A = 5 × 531363, B = 3 × 531345, C = 5313105, D = 531379. □
Therefore, we have found that the solutions the indefinite equations as (14) and (18) or more.
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.