Final Answer:
The statement is incorrect. It is possible for d to equal n under the given conditions.
Step-by-step explanation:
Consider the given equation x² ≡ y (mod n), where x ≠ y (mod n) and x ≠ -y (mod n). This implies that x and y are distinct residues modulo n. Let d = gcd(x - y, n). The condition x ≠ -y (mod n) means x + y ≠ 0 (mod n), and since x ≠ y (mod n), we have x - y ≠ 0 (mod n). Therefore, x - y and n are relatively prime.
Now, consider d = gcd(x - y, n). Since x - y and n are relatively prime, d must be 1, as gcd(a, b) = 1 when a and b are relatively prime. Thus, d can indeed equal n if x - y and n are coprime.
In conclusion, the initial assertion that d cannot equal n is false. Under the given conditions, it is possible for d to equal n when x - y and n are relatively prime