Let  be the th Bernoulli number and consider

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.

REFERENCES:

Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgensohn, R. "Giuga's Conjecture on Primality." Amer. Math. Monthly 103, 40-50, 1996.

Kellner, B. C. Über irreguläre Paare höherer Ordnungen. Diplomarbeit. Göttingen, Germany: Mathematischen Institut der Georg August Universität zu Göttingen, 2002. https://www.bernoulli.org/~bk/irrpairord.pdf.

Kellner, B. C. "The Equivalence of Giuga's and Agoh's Conjectures." 15 Sep 2004. https://arxiv.org/abs/math.NT/0409259.

Sloane, N. J. A. Sequence A046094 in "The On-Line Encyclopedia of Integer Sequences."