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I've been messing around with rising factorials and found something odd.

$\frac{x^{(n)} - d^{(n)}}{n!}$

seems to never be prime when $n = 4$ for all $x$ and $0 < d < x$ (I've been looking for months for a counter example but have't found any). Moreover, for other cases of $n > 1$, it seems to equal to prime numbers only when $(x - d)$ divides $n!$

Is it true for $n=4$? Is there a reason for the other cases? I feel like I missed something obvious.

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    Where $x^{(n)} = x(x+1)(x+2)...(x+n-1)$ – 2017-01-10
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    FWIW, $\frac{x^{(4)}-d^{(4)}}{4!}=\frac{1}{24} (x-d) (d+x+3) \left(d^2+3 d+x^2+3 x+2\right)$. – 2017-01-10

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