The Discussion On The Problems Of Two Kinds Of Diophantine Equations

The Diophantine equation is also called the indefinite equation.It is an important branch of number theory,and it has a close relationship with discrete mathematics.In this paper,we mainly study two types of Diophantine equations,the specific contents are as follows:In the first chapter,we give an introduction,a brief introduction to the Diophantine equation and the Euler function.In the second chapter,we give preliminary knowledge,mainly some properties and theorems of Pell equation and Euler function.In the third chapter,by using the method of recursive sequence and some properties of the solutions to Pell equation,we discuss the existence of the integer solution of the Diophantine equation x2-22y2=1 and y2-Dz2=1764 and get the following conclusions:If D =2p1…ps,1?s?4(p1,…,ps are distinct odd primes),then(i)it has only trivial solutions(x,y,z)=(±197,±42,0)with the exception that D=2×77617;(ii)it has integer solutions(x,y,z)=(±30580901,±6519870,±16548),(±197,±42,0)where D=2x77617;if D=pm(m? Z+,pare primes),then it has only trivial solutions(x,y,z)=(±197,±42,0).In the fourth chapter,by using the properties of the Euler functions,we discuss the solvability of the compound Euler functionequation ?(n-?(n-?(?(n))))=2t(t?N)and get the following conclusions:The upper bound of positive integer solutions n is 32768t8;if t=1,then it has positive integer solutions n=5,7,8,9,10;if t=2,then it has positive integer solutions n=11,12,14,15,16,18,20,21;if t=3,then it has positive integer solutions n=13,19,25,30,31,39,45;if t=4,then it has positive integer solutions n=22,24,26,27,28,32,36,40,42.