The Research On Solvability Problems Of Several Indeterminate Equations

Indeterminate equations refer to the equations that the number of unknowns is greater than the number of equations,and its solution is positive integer.It has a long history in number theory.Its research results not only play an important role in the branch of mathematics,but also have lots of application value in the non mathematics subject.In this paper,we study some special indeterminate equations by using the method of elementary number theory.I have gotten some positive integer solutions of them.Firstly,we study the integer solution of linear indeterminate equation in eight unknowns and discuss the number of solutions.By using some elementary number theory methods,we construct two indeterminate equations in forth unknowns and one indeterminate equations in two unknowns.Finally,we obtained all the integer solutions and the number of solutions of this equation.Secondly,we discuss the solution of indeterminate equation x3 ± 8 = 2pqy2.Let p,q be fixed odd integer with p ≡ 1(mod 24),q = 12s2+1(s be odd positive integer),(p/q)=-1.By using some elementary number theory methods,we prove that(1):the equation x3 + 8 = 2pqy2 only have integer solution(x,y)=(-2,0)with 3|(2n+1).(2)the equationx3-8 = 2pqy2 does not have the positive integer solution with gcd(x,y)= 1.Thirdly,we discuss the solution of indeterminate equation x3 +73=l4y2.By using some elementary number theory methods,we prove that the equation x3+73 = 14y2 has only the positive integer solution(x,y)=(7,7)and(x,y)=(161,546).Fourthly,we study the integer solutions of the Jesmanowicz conjecture on Pythagorean triples.By using some elementary number theory methods,we obtain and prove two new results for the conjecture and generalize some results of references[51-55].Fifthly,We study the solution of the exponential indeterminate equation(4k)x+by =(b+4k)z,·By using some elementary number theory methods,some known results on exponential indeterminate equations and an extension of the stormer theorem on Pell’s equations,we prove the equation(4k)x+by =(b+4k)z has only the positive integer solution(x,y,z)=(1,1,1)and generalize some results of references[61].Finally,we conclude the solutions of indeterminate equations and particular indeterminate equations and point out some issues which need further research.