This Liouville's theorem (the most unknown of his work) : "If $n \in \mathbb{N^{*}}$ and $p>5$ a prime number, then the equation $(p-1)! + 1 = p^{n}$ has no solution."
The standard proof is clear. But I was wondering if there are examples of this theorem other than arithmetic. I mean does it only prove that the equation of the form : $(p-1)! + 1 = p^{n}$ has no solution or are there other applications ?
Thanks in advance.
The meaning of the theorem is this: if $n$ is a natural number ($1,2,3,\cdots$), and $p$ is a prime number other than $2,3,5$, then $(p-1)! + 1$ and $p^n$ are not equal.
To find "examples", pick any natural number and any prime number larger than $5$, e.g.:
$$(11-1)!+1 \neq 11^4$$
$$(7-1)!+1\neq 7^3$$
$$((2^{57885161}-1)-1)! + 1 \neq (2^{57885161}-1)^2$$
And so on.
It is hard to know what you are asking for regarding the "meaning" of this theorem. It is simply the statement that two quantities can never be equal, regardless of what we plug in for $n$ and $p$. Philosophically, it resembles a statement like "an odd number can never be equal to an even number."
As such, it is hard to give good examples or illustrations—how can you demonstrate something that is impossible?—but if you are looking for a proof, why not read what Liouville himself wrote on the subject?
The result you are looking for does not appear to have been published in 1844 (Liouville appears to have been working on diophantine approximations at the time), but rather 1856, in Journal de Mathématiques Pures et Appliquées. There is a citation in this 1996 paper, but you are unlikely to find an obscure paper from 1856 without good library access (sciencedirect's archives only go back to 1997, for example).
You can also read the paper of Yu and Liu that I linked in the last paragraph. They discuss a generalization of this equation proposed by Erdös and Graham in 1979. The 1979 paper, "Old and New Problems and Results in Combinatorial Number Theory", can easily be found with a Google search—it is much longer, but contains a lot of exposition and may shed a little light on why mathematicians continue to be interested in this equation.