Agoh's Conjecture

Let B_k be the kth Bernoulli number and consider

 nB_(n-1)=-1 (mod n),

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 -1 iff n is prime. There are no counterexamples less than n=49999 (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 n=1, 2, ..., the minimal integer residues nB_(n-1) (mod n) is 0, -1, -1, 0, -1, 0, -1, 0, -3, 0, -1, ... (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.

See also

Bernoulli Number, Congruence, Giuga's Conjecture, Minimal Residue

Explore with Wolfram|Alpha


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., B. C. "The Equivalence of Giuga's and Agoh's Conjectures." 15 Sep 2004., N. J. A. Sequence A046094 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Agoh's Conjecture

Cite this as:

Weisstein, Eric W. "Agoh's Conjecture." From MathWorld--A Wolfram Web Resource.

Subject classifications