Giuga's Conjecture on Prime Characterization

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an integer $n\geq2$, put $S(n)=\sum_{a=1}^{n-1}a^{n-1}$. Divisibility $n\mid S(n)+1$ means $S(n)\equiv-1\pmod n$. A prime integer is an integer greater than one with no positive divisors other than one and itself.

Statement recorded in the supplied source.

Is the equivalence \[\forall n\geq2,\qquad n\mid\left(1+\sum_{a=1}^{n-1}a^{n-1}\right)\quad\Longleftrightarrow\quad n\text{ is prime}\] true? In particular, does the displayed divisibility condition rule out every composite integer? [S2]

Short English statement

Does the power-sum congruence characterize primes, with no composite integer satisfying it?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1870 (GUY-A17). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bernd C. Kellner, The Equivalence of Giuga’s and Agoh’s Conjectures, arXiv:math/0409259 (2004). Section1 and Conjectures2.1--2.2, pages1--2. https://arxiv.org/pdf/math/0409259.