3 Let such that . Show that the equation has no integer solution.

**Sketch of the Solution:**

**Claim 1: **There cannot be a negative integer solution. Suppose other wise. If possible x= -k (k positive) be a solution.

Then we have . Clearly this is impossible as and .

**Claim 2:** 0 is not a solution (why?)

**Claim 3:** There cannot be a positive integer solution. Suppose other wise. If possible x=k (k positive) be a solution.

Then we have

This implies that the right hand side is divisible by k which again implies that d is divisible by k (why?).

Let d=d’k

Now

Thus

Hence the equality