On Two Special Cases Of The Diophantine Equation X~2+c=y~n

In this paper, we discuss two special cases of the Diophantine Equation x2+c=yn, i. e.x2+q2k+1=yn and x2+5a17b=yn.In the first part, it has been proved that if q is an odd prime, q=7 (mod 8), n is an odd integer≥5, n is not a multiple of 3 and (h, n)=1, where h is the class number of the filed Q((?)),then the diophantine equation x2+q2k+1=yn has no solutions (q, n, k, x,y) with y odd.In the second part, we find all the solutions of the Diophantine equation x2+5a17b=yn in positive integers x,y,a,b,n≥3 with x and y coprime.