05AB1E, 15 14 bytes

∞+.ΔαD3mαy¿yp+

Inspired by @Giuseppe's Gaia answer, so make sure to upvote him as well!

Try it online or verify all test cases.

Explanation:

∞            # Push an infinite positive list: [1,2,3,...]
 +           # Add the (implicit) input `n` to each: [n+1,n+2,n+3,...]
  .Δ         # Find the first `n+b` which is truthy for:
    α        #  Take the absolute difference with the (implicit) input: [1,2,3,...]
     D       #  Duplicate this `b`
      3m     #  Cube it: b³
        α    #  Take the absolute difference with the `b` we've duplicated: |b-b³|
         y¿  #  Get the greatest common divisor with the current `b+n`: gcd(|b-b³|,b+n)
    yp       #  Check whether `b+n` is a prime number (1 if prime; 0 if not)
    +        #  Add them together: gcd(|b-b³|,b+n)+isPrime(b+n)
             #  (NOTE: only 1 is truthy in 05AB1E)
             # (after which the result is output implicitly as result)