Any composite number 
with 
for all prime divisors 
of 
. 
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 have sum minus product equal to 1, and any Giuga number of higher order must have at least 59 factors. The smallest odd Giuga number must have at least nine prime factors.
,
Pseudoperfect Numbers, and Partially Weighted Graphs." Math. Comput. 69,
407-420, 1999.Kellner, B. C. Über irreguläre Paare
höherer Ordnungen. Diplomarbeit. Göttingen, Germany: Mathematischen
Institut der Georg August Universität zu Göttingen, 2002.