Indeterminate Equation Of X~2+K=Y~n Rational Integer Solutions

In number theory, indeterminate equation very characteristics, has been causingthe attention and study of numerous mathematical enthusiasts. Not only does it developpowerfully, but also it is widely used in economics, discrete mathematics, physics andother fields. We have a consensus: there is much work to be likely to reach someDiophantine equation up to be addressed, but particularly difficult practical andtheoretical problems. Therefore, many scholars at home and abroad have researched itfrom its extensive and depth. It is an important question about the Diophantine Equationx²C Bynof rational integer solution in number theory. When C=1, B=1,Lebesgue proved that the equation has no rational integer solutions. In1842, CatalanConjecture made a suppose that x^m-yⁿ=1（x, y,m,n Z,x,y,m,n＞1） has only a soluti-on32231.This problem is resolved by Ke Zhao completely in1962.This pape mainly used the Pell equation, elementary methods, the nature ofalgebraicnumber theory, congruence and other methods to research indeterminate equationX2K Ynas follows.First，it discussed indeterminate equationx²±1=ky³’s the integer solutions whenk3,4,k e21e∈Z andx²k²y³’s the integer solutions whenk=p=1（mod4）,k p3m od4:1. The equationx²+1=3y³has no integer solutions;2. The equationx²-1=3y³only has integer solution （±1,0）,（±2,1）and （±5,2）；3. The equationx²+1=4y³has no integer solutions; 4. The equationx²+1=ky³only has integer solution （x±e, y=1）;5. The equationx²+k²=y³has integer solutionx=（±e³+e）， y=e2+1; whenk=e2=1.6. The equationx²+k²y³has integer solution whenk p3（mod4）.Second，it discussed the integer solutions of the indeterminate equationx²±p=y³andx²+9=y⁵，（p=-5,±13,17）.1. The equationx²*5=y³only has integer solutions （±2,-1）;2. The equationx²*13=y³has no integer solutions;3. The equationx²+13=y³only has integer solutions （±70,17）;4. The equationx²+17=y³has no integer solutions;5. The equationx²+4=y¹¹has no integer solutions.Third，it discussed the rational integer solutions of indeterminate equationx²+4=yⁿ1. The equationx²4ynhas no integers solution when n4and even oddnumbers;2. The equationx²+4=y⁹has no rational integer solution;3. The equation x²+4=y¹¹ has no rational integer solution.