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In this paper we shall discuss the integral solutions of the diophantine equation Y 2 +K=X 5 , where AT is a square-free positive integer. We shall prove the following:THEOREM. If the class number h of the quadratic field Q(j-K) is not divisible by 5, and if K^ 8L-1, then the equation Y 2 +K=X 5 has no nonzero integral solutions with the exceptions of K= 19, 341.In these cases the solutions will be as follows :(22434) 2 + 19 = (55) 5 , (275964Ó) 2 + 341 = (377) 5 . D The ideal equation [Y+J-K]• [Y-J-K\=X 5 leads to finitely many equations [see e.g. [3]] of the form f (A, B)=m, where ƒ is a homogeneous polynomial of degree 5.The case Y+

We show that if the class number of the quadratic field A(\/-fc) is not 2 divisible by 5, and if fc is not congruent to 7 modulo 8, then the equation Y + k = JT has no solutions in rational integers X, y with the exception of k = 1, 19, 341. In this paper we shall discuss the solutions in integers of the diophantine equation (1) Y2+k = Xs, where k is a positive square-free integer. We shall show that if the class number of the quadratic field Q{ y/-k) is not divisible by 5, and if k is not congruent to 7 modulo 8, then Eq. (1) has no solutions with three exceptions. We will also find all the solutions of Eq. (1) in the exceptional cases. Results of this note are announced in [6]. Proof. Any odd prime P dividing Cx divides y/-~k ; hence P ramifies, and so we have P = [p]2 where p is a prime number. For such P, it is clear that a{V) = P. If P is a prime dividing 2, then P ramifies if k ^ 3 (mod 8) and 2 remains prime if