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I just read this equation, which is surprisingly true. The interesting equation turned me to find all the positive integer solutions $x$, $y$, and $z$ s.t. $x - \frac{y}{z} = p!$, where $p = \frac{x - y}{z}$ is also an integer.

Note that, in the case of $40 - 32/2 = 4!$ we have $x = 40,~y = 32,~z = 2,~p = \frac{x-y}{z} = 4$.

I wonder if there exist any analytical solution to such factorial polynomials.

Mark McClure
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Johnny Chien
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1 Answers1

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Writing $y$ in terms of the integers $p$, $z$, and $x$ we have $y=x-pz$ and $$x-\frac{x}{z}+p=p!$$ Since all other terms are integer, so must be $\frac{x}{z}$. Let $x=kz, k\in\mathbb{Z}$.

Transforming the inital eqation we get $k(z-1)=p!-p$. Now for finding all solutions, you can iterate over $p$ and choose $k$ from the divisors of $p!-p$.

Given $p$ and $k$ you get $z=1+\frac{p!-p}{k}$, $x=kz$, $y=x-pz$.

Note that when $k<p$ then $y$ is negative and when $k=p$ it is zero.

$p=4$ and $k=20$ leads to your example.