不定方程的解（2）

1、Solutions of Indefinite Equations(2)Zengyong LiangMCHH of Guangxi, Nanning, ChinaAbstractIndefinite equation is an unsolved problem in number theory. Through exploration,the author has been able to use a simple elementary algebraic methodto solve the solutions of all three variable indefinite equati

2、ons. In thispaper, we will introduce and prove the solutions of Pythagorean equation,Fermats theorem, Bill equation and so on.Subsections 5 and 6 are supplemented here.KeywordsIndefinite Equation, Fermats Last Theorem, Algebraic Transformation,L-Algorithm5 Summary5.1 L-algorithmThe specific steps of

3、 L-algorithm (three-step method) are as follows:1) First, find out the original equation model which is lower than the original equation. (or a new equation is formed by the sum value L (f) of the left term of the equation, and the unknown number of the equation is set to a smaller value), as shown

4、in the following equation: L(f)= w. For example, suppose the original equation has three termsax + by =cu, (21) then L(f)= ax + by =w.2) Using the principle that both sides of the equation are multiplied by an integer at the same time, a new higher order equation similar to the proposition is genera

5、ted( ax + by ) wxy =w wxy (22) Determine the solution of each unknown number of the equation: We write (22) as the following expression (imitating the form required by equation (23) in the proposition) (awy) x +( bwx)y =wu (23)where u|xy+1. Now, we can determine a= awy , b= bwx , c=w.6 Higher Order

6、Indefinite Equation With CoefficientsSuppose there is a problem, find3a5+5b4=c3 (24)You can find the solution first w=3×35+5×543854 , then wxy =385420 , and385420× 36+385420×55 = 3854×3854203× (38544×3)5+ 5×(38545×5)4 = 3854×385420 . So a=38544×

7、3 ,b=38545×5, c=38547 .In this way, the solution of the equation with coefficients is found. For higher-order indefinite equations with coefficients with more terms, the solutions canYou can find the solution first w=3×35+5×543854 , then wxy =385420 , and385420× 36+385420×55

8、 = 3854×3854203× (38544×3)5+ 5×(38545×5)4 = 3854×385420 . So a=38544×3 ,b=38545×5, c=38547 .In this way, the solution of the equation with coefficients is found. For higher-order indefinite equations with coefficients with more terms, the solutions can also be

9、 found. .For example this equation is 11a2+7b3+5c4 = 3d5 (25)Find common factor wxy 11×32+7×33+5×34= 99+135+405=639=3*60=3×3×71Hence the common factor is (3×7)24 , then(3×7)24 ×(11×32+7×33+5×34) =3×3×71× (3×7)2411×(313&#

10、215;7112) 2+7×(39×718) 3 +5×(37×71 6)4 =3*(3*715)5So a=313×7112 ,b=39×718, c=37×71 6, d=3*715 .Generally, there isk1ax+k2by+ k3cz = k4d u (26)Obviously, u and x, y, z are mutually prime, and there is a solution using the l-algorithm Using this method flexibly, more

11、 types of higher-order indefinite equations can be solved.6 Analysis and Discussion6.1 Behr and swinnestone Dale conjectureBach and swinnestone Dale conjectured: "mathematicians are always fascinated by the characterization of all integer solutions of algebraic equations such as x 2 + y 2 = Z 2

12、. Euclid once gave a complete solution to this equation, but for more complex equations, it becomes extremely difficult for more complex equations. In fact, as matiyasevich pointed out, Hilbert's tenth problem is insoluble, that is, there is no general method to determine whether such a method has an integer solution.”7. Now, we have been able to find all integer solutions to any algebraic equation of the from a x + by = cz .It is proved that there is no integer solution for the equation without solution. Then, we solve the problem proposed b