Giuga's criterion for the congruence $\sum_{i<n} i^{n-1} \equiv -1 \pmod n$

Giuga's criterion. Let $n \ge 2$ be an integer. Then the Giuga congruence

holds if and only if for every prime $p$ dividing $n$ one has

The first family of conditions says that $n$ is a Giuga number (in the usual sense, extended here to prime $n$, for which $n/p - 1 = 0$ and the conditions are vacuous); the second family is Korselt's criterion in its $n/p$ form, and together with the first it says that $n$ is a Carmichael number. The criterion is the arithmetic heart of the Agoh--Giuga conjecture: it converts the congruence, a statement about a sum of $n-1$ large powers, into finitely many divisibility conditions on the prime factorisation of $n$, and it is the reason a counterexample to the conjecture is known to be simultaneously a Carmichael number and a Giuga number.

Note that the first condition forces $n$ to be squarefree: if $p^2 \mid n$ then $p \mid n/p$, which is incompatible with $p \mid n/p - 1$.

Formalization Note The sum is taken over the integers $i$ with $0 < i < n$, and all arithmetic is in the natural numbers, so the congruence $\sum i^{n-1} \equiv -1 \pmod n$ is written in the equivalent divisibility form $n \mid 1 + \sum_{i=1}^{n-1} i^{n-1}$. The subtractions $n/p - 1$ and $p - 1$ are truncated natural subtraction, which is harmless because $p \ge 2$ and $n/p \ge 1$ for every prime divisor $p$ of $n$.