prove that $\forall k\geqslant1\land k\in\mathbb{Z}.\nexists q\in\mathbb{Z}.q^{2}=(k-i)(k+i)$

Question

prove that $\forall k\geqslant1\land k\in\mathbb{Z}.\nexists q\in\mathbb{Z}.q^{2}=(k-i)(k+i)$. My try: use gaussian integer to claim that we have ufd. the problem, is that k might not be irreducible in gaussian primes. Edit: without using that non zero squares cannot differ by 1.

Answer 1

Hint: Two nonzero perfect squares in $\Bbb{Z}$ cannot differ by $1$.