Giuga Number

Any Composite Number $n$ with $p\vert(n/p-1)$ for all Prime Divisors $p$ of $n$. $n$ is a Giuga number Iff

$$\sum_{k=1}^{n-1} k^{\phi(n)} \equiv -1{\rm\ (mod\ }n)$$

where $\phi$ is the Totient Function and Iff

$$\sum_{p\vert n} {1\over p}-\prod_{p\vert n} {1\over p}\in\Bbb{N}.$$

$n$ is a Giuga number Iff

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

where $B_k$ is a Bernoulli Number and $\phi$ is the Totient Function. Every counterexample to Giuga's conjecture is a contradiction to Argoh'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), ... (Sloane's 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.

See also Argoh's Conjecture, Bernoulli Number, Totient Function

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 A007850 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.