Some Study Of The Diophantine Equation Of The Type1+X=Y+Z

Diophantine equation is an important subject in number theory and closely connected with algebraic number theory, combinatorics, algebraic geometry and computer science etc. The results in Diophantine equation play an important role both in some branches of mathematics and in other subjects, such as economics, physics. So there are still many people who have great interested in Diophantine equation. The exponential Diophantine equation1+X=Y+Z (XYZ=pαqβrγ)(p,q,r being distinct prime numbers and α,β,γ being non-negative integers) can be used to research the classification of the finite simple group. In1985, Alex L J. got the all non-negative integer solutions when {p、q、r}={2、3、5} and {p、r}={2、3、7}. In this paper, with the help of computer and the results of advanced methods several kinds of exponential Diophantine equation1+X=Y+Z were studied by elementary methods. The main results are as follows:1、Obtaining the all solutions in non-negative integers of the following equations1+2x=5yllz+2u5v11w,1+5x=2y11z+2u5v11w1+11x=2y5Z+2u5v11w,1+5x11y-2z5u+2v11w2、Proving two special cases of the exponential Diophantine equations1+13x=2y19z+2u13v19w,1+13x37y=2z13u+2v37w having only trivial solution.