mariofelixpasku wrote:If 17! + 2 ≤ K ≤ 17! + 17, is K ever a prime number?

The quick answer is no

First of all 17! + 2 is not prime. Here's why:
17! + 2 = (17)(16)(15)....(3)(2)(1) + 2
Notice we can factor out a 2 here to get:
17! + 2 = 2[(17)(16)(15)....(3)(1) + 1]
This means that 17! + 2 is divisible by 2, which means it is not prime

Similarly, 17! + 3 is not prime.
17! + 3 = (17)(16)(15)....(4)(3)(2)(1) + 3
Notice we can factor out a 3 here to get:
17! + 3 = 3[(17)(16)(15)....(4)(2)(1) + 1]
This means that 17! + 3 is divisible by 3, which means it is not prime

As you can see, we can continue the process for 17!+4, 17!+5,... 17!+17

Cheers,
Brent