Giuga's Conjecture

If $n>1$ and

$$n\vert 1^{n-1}+2^{n-1}+\ldots+(n-1)^{n-1}+1,$$

is $n$ necessarily a Prime? In other words, defining

$$s_n\equiv \sum_{k=1}^{n-1} k^{n-1},$$

does there exist a Composite $n$ such that $s_n\equiv -1\ \left({{\rm mod\ } {n}}\right)$? It is known that $s_n\equiv -1\ \left({{\rm mod\ } {n}}\right)$ Iff for each prime divisor $p$ of $n$, $(p-1)\vert(n/p-1)$ and $p\vert(n/p-1)$ (Giuga 1950, Borwein et al. 1996); therefore, any counterexample must be Squarefree. A composite Integer $n$ satisfies $s_n\equiv -1\ \left({{\rm mod\ } {n}}\right)$ Iff it is both a Carmichael Number and a Giuga Number. Giuga showed that there are no exceptions to the conjecture up to $10^{1000}$. This was later improved to $10^{1700}$ (Bedocchi 1985) and $10^{13800}$ (Borwein et al. 1996).

See also Argoh's Conjecture

References

Bedocchi, E. ``The $\Bbb{Z}(\sqrt{14}\,)$ Ring and the Euclidean Algorithm.'' Manuscripta Math. 53, 199-216, 1985.

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

Giuga, G. ``Su una presumibile propertietà caratteristica dei numeri primi.'' Ist. Lombardo Sci. Lett. Rend. A 83, 511-528, 1950.

Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 20-21, 1989.