A simple Diophantine equation

Let’s solve a Diophantine equation together using only elementary considerations. Suppose we are asked to determine all the integers X and Y such that X³-X+1=3Y⁵. As is usually done, we assume that such a solution exists, let’s say (X,Y), and we try to understand if we can make the value of X and Y explicit or find a contradiction that allows us to state that the equation has no solutions. Let’s start. We observe that 3 is a divisor of X³-X+1 given that X³-X+1=3Y⁵. Now, we have 3 possibilities: if I divide X by 3 I get remainder 0, if I divide X by 3 I get remainder 1 or if I divide X by 3 I get remainder 2. The first case is completely absurd because if X=3k for some integer k then I would find 27k³-3k+1=3Y⁵ and therefore 3 divides 1, absurd. The second is also absurd because if X=3k+1 with k in Z we get (3k+1)³-3k=3Y⁵. Developing the left-hand side we still find that 3 divides 1, absurd. But then X=3k+2 with k in Z and therefore (3k+2)³-3k-1=3. Therefore 27k³+54k²+36k-7=3 it follows that 3 divides 7, absurd since 7 is prime. Therefore there are no solutions.