If you're not into "hunting for primes", here is a slight and concrete variant.
Let $a-2=(k+2)!$ and $b+2=(k+2)!$. Then proceed in the same manner as Fedor's example: $$\text{gcd}(a+i,b-i)=\text{gcd}((a-2)+i+2,b+2-(i+2))>1.$$
BTW, very cool drawing, Joseph!