Question

there are no x, y ∈ Z + such that x 2 + x + 1 = y 2 . (Hint: what is the smallest square integer that is larger than x 2?)

Answer 1

To prove that there are no values of x and y that satisfy the equation x^2 + x + 1 = y^2, we can consider the smallest square integer that is larger than x^2, which is (x+1)^2.

Step-by-step explanation:

To prove that there are no values of x and y that satisfy the equation x^2 + x + 1 = y^2, we can consider the smallest square integer that is larger than x^2, which is (x+1)^2. If (x+1)^2 is a perfect square, it means that there is no integer value of y that can make the equation true. Let's write out the steps:

1. Start with the equation x^2 + x + 1 = y^2
2. Consider the smallest square integer larger than x^2, which is (x+1)^2
3. Expand (x+1)^2 to get x^2 + 2x + 1
4. Now we compare x^2 + x + 1 with x^2 + 2x + 1
5. If x^2 + x + 1 is not equal to (x+1)^2, then there are no values of x and y that satisfy the equation