Argoh's Conjecture

Let $B_k$ be the $k$th Bernoulli Number. Then does

$$nB_{n-1}\equiv -1\ \left({{\rm mod\ } {n}}\right)$$

Iff $n$ is Prime? For example, for $n=1$, 2, ..., $nB_{n-1}$ (mod $n$) is 0, $-1$, $-1$, 0, $-1$, 0, $-1$, 0, $-3$, 0, $-1$, ... (Sloane's A046094). There are no counterexamples less than $n=5,600$. Any counterexample to Argoh's conjecture would be a contradiction to Giuga's Conjecture, and vice versa.

See also Bernoulli Number, Giuga's Conjecture

References

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

Sloane, N. J. A. Sequence A046094 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.