a) José learned a method for calculating the product of two numbers more quickly based on factorization:

$$
(n-k)(n+k)=n^{2}-k^{2}
$$

To calculate 23 $\cdot 17$, he chooses $n=20, k=3$ and calculates:

$$
23 \cdot 17=20^{2}-3^{2}=400-9=391
$$

Determine, without using a calculator, the value of $\sqrt{1001 \cdot 1003+1}$.

b) Verify that $(n(n+3)+1)^{2}=n(n+1)(n+2)(n+3)+1$.

c) Determine, without using a calculator, the value of:

$$
\sqrt{(2014)(2015)(2016)(2017)+1}
$$