Giuga's conjecture in arithmetic form: no composite Giuga--Carmichael number

Giuga's conjecture, arithmetic form. Let $n \ge 2$ be an integer and suppose that for every prime $p$ dividing $n$ both

hold. Then $n$ is prime.

Equivalently: no composite number is simultaneously a Giuga number and a Carmichael number. This is the open kernel of the Agoh--Giuga conjecture. By Giuga's criterion, an integer $n \ge 2$ satisfies $\sum_{i=1}^{n-1} i^{\,n-1} \equiv -1 \pmod n$ exactly when the two displayed families of divisibilities hold, so the conjecture "the congruence forces primality" is equivalent to the statement above. A composite $n$ satisfying the hypotheses is known to have at least $13{,}800$ digits (Borwein, Borwein, Borwein and Girgensohn, 1996), and later computations push the bound much further.

Primes satisfy the hypotheses trivially: for $n = p$ prime the only prime divisor is $p$ itself, $n/p - 1 = 0$, and every number divides $0$.

Formalization Note All arithmetic is in the natural numbers; $n/p$ is exact division since $p \mid n$, and the subtractions are truncated natural subtraction, which is harmless because $p \ge 2$ and $n/p \ge 1$.