A Giuga number is a composite number 
with 
for all prime factors 
of 
. A composite number 
is a Giuga number iff
 |
(1)
|
where 
is the totient function and iff
 |
(2)
|
is a Giuga number iff
 |
(3)
|
where 
is a Bernoulli number and 
is the totient function.
Every counterexample to Giuga's conjecture is a contradiction to Agoh's
conjecture and vice versa. The smallest known Giuga numbers are 30 (3 factors),
858, 1722 (4 factors), 66198 (5 factors), 2214408306, 24423128562 (6 factors), 432749205173838,
14737133470010574, 550843391309130318 (7 factors),
244197000982499715087866346, 554079914617070801288578559178
(8 factors), ... (OEIS A007850).
It is not known if there are an infinite number of Giuga numbers. All the above numbers satisfy 
.
A Giuga number satisfying 
must have at least 59 prime factors. Any odd
Giuga number must have at least nine prime factors.
,
Pseudoperfect Numbers, and Perfectly Weighted Graphs." Math. Comput. 69,
407-420, 2000.