Several Studies On Index Diophantine Equation Problems

Diophantine equations,also known as indeterminate equations.It is a very important branch of number theory,and it is a very hot research topic.Recently index Dio-phantine equations are very active.because it not only has been very developed but also be fully applied to various other areas.With the exponential Diophantine equation itself conti-nuing to evolve,its research results play an important role on other branches of mathematics and many non-mathematics subjects.Therefore,many scholars have conducted extensive studies.In this paper,I have studied several exponential Diophantine equations by using defin-itions and nature of congruence,quadratic residue,the Pell equation,Legendre symbol,Jaco-bi symbol and so on.The key elements are as the follows:1.The second chapter mainly introduces the research progress of exponential Diophantine equations,and uses the way of the quadratic residue.I have studied the solution of the Diophantine equation(an-1)(bn-1)= x2 when(a,b)=(5k1 + 3,5k2 +2),and completely solved the class of Diophantine equations when k1,k2 satisfy some conditions.2.Using the elementary methods,I study the solution of the Diophantine equation(8n-1)(bn-1)= x2,3 | b and I have discussed the solution of(2Pn-1)(bn-1)= x2 when p is a prime.3.I have discussed the solvability of(an-1)((a + 1)n-1)= x2.4.I give a note to the equation ax +by = z2 by using the elementary methods.5.Summary of this article on solving the exponential Diophantine equations and some issues which need further research are pointed out.