If and
is necessarily a prime? In other words, defining
does there exist a composite such that ? It is known that iff for each prime divisor of , and (Giuga 1950, Borwein et al. 1996); therefore, any counterexample must be squarefree. A composite integer satisfies iff it is both a Carmichael number and a Giuga number. Giuga showed that there are no exceptions to the conjecture up to . This was later improved to (Bedocchi 1985) and (Borwein et al. 1996).
Kellner (2002) provided a short proof of the equivalence of Giuga's and Agoh's conjectures. The combined conjecture can be described by a sum of fractions.