Consider a sequence of polynomials with real coefficients defined by
$$p_{0}(x) = \left(x^{2}+1\right)\left(x^{2}+2\right) \cdots\left(x^{2}+1009\right)$$
with subsequent polynomials defined by $p_{k+1}(x) := p_{k}(x+1) - p_{k}(x)$ for $k \geq 0$. Find the least $n$ such that
$$p_{n}(1) = p_{n}(2) = \cdots = p_{n}(5000)$$