where the residues of fractions are taken in the usual way so as to yield integers, for which the minimal residue is taken. Agoh's
conjecture states that this quantity is iff is prime. There are no counterexamples
less than
(S. Plouffe, pers. comm., Jan. 28, 2003). Any counterexample to Agoh's
conjecture would be a contradiction to Giuga's conjecture,
and vice versa.
For ,
2, ..., the minimal integer residues (mod ) is 0, , , 0, , 0, , 0, , 0, , ... (OEIS A046094).
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.
Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgensohn, R. "Giuga's Conjecture on Primality." Amer. Math. Monthly103,
40-50, 1996.Kellner, B. C. Über irreguläre Paare höherer
Ordnungen. Diplomarbeit. Göttingen, Germany: Mathematischen Institut der
Georg August Universität zu Göttingen, 2002. http://www.bernoulli.org/~bk/irrpairord.pdf.Kellner,
B. C. "The Equivalence of Giuga's and Agoh's Conjectures." 15 Sep
2004. http://arxiv.org/abs/math.NT/0409259.Sloane,
N. J. A. Sequence A046094 in "The
On-Line Encyclopedia of Integer Sequences."
be the 
th
Bernoulli number and consider
iff 
is prime. There are no counterexamples
less than 
(S. Plouffe, pers. comm., Jan. 28, 2003). Any counterexample to Agoh's
conjecture would be a contradiction to Giuga's conjecture,
and vice versa.
,
2, ..., the minimal integer residues 
(mod 
) is 0, 
, 
, 0, 
, 0, 
, 0, 
, 0, 
, ... (OEIS A046094).
