| Another name of indeterminate equations are Diophantine equations. Indeterminate equation is a very important branch and a very hot research topic in the number theory. The achievements of indeterminate equations play an important role both in other branch of mathematics and in other subjects. With the own development of indeterminate equations, there are many scholars who do broadly and deeply research in them.On this thesis, by using congruence, quadratic residue, the nature of Pell equation, Gauss quadratic reciprocity law, Legendre symbol definition and properties, recurrence sequence, discriminant method of Euler, method of Baker, etc. four types of solvability problem of special form of indeterminate equations are studied. The main contents of this thesis are as following:First, the problem of integer solution of the indeterminate equations x3±C=Dy2 are discussed. First of all, it is proved that the indeterminate equation x3-33=3pqy2 only has integer solution (x, y)=(3,0) and the indeterminate equation x3+33=3pqy2 only has integer solution (x, y)=(-3,0). Then, it is proved that the indeterminate equation x3+23=3pqy2 has not integer solution for (x, y)=1. Finally, it is proved that the indeterminate equation x3-23=pqy2 only has integer solution (x, y)=(2,0)for(x, y)=1.Second, the problem of integer solution of the indeterminate equations x2+C=y3 are discussed. It is proved that the indeterminate equation x2+260642=y3 only has integer solution (x, y)=(±1265,123).Third, the problem of integer solution of the exponential indeterminate equations (na)x+(nb)y=(nc)z are discussed. It is proved that the indeterminate equation (285n)x+ (68n)y=(293n)z only has positive integer solution (x, y, z)=(2,2,2).Forth, the solvability problem of the indeterminate system of equations are studied. All positive integer solution of the indeterminate system of equations and the upper bound of them are concluded, when m=10. Based on it, it has been generalized to more general upper bound of the positive integer solution of indeterminate system of equations... |
