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.
Ring and the Euclidean Algorithm." Manuscripta
Math. 53, 199-216, 1985.Borwein, D.; Borwein, J. M.;
Borwein, P. B.; and Girgensohn, R. "Giuga's Conjecture on Primality."
Amer. Math. Monthly 103, 40-50, 1996.Giuga, G. "Su
una presumibile propertietà caratteristica dei numeri primi." Ist.
Lombardo Sci. Lett. Rend. A 83, 511-528, 1950.Kellner, B. C.
Über irreguläre Paare höherer Ordnungen. Diplomarbeit. Göttingen,
Germany: Mathematischen Institut der Georg August Universität zu Göttingen,
2002.