Giuga's Conjecture on Primality

Definitions and mathematical statement

For an integer $n\ge2$, put $S(n)=\sum_{a=1}^{n-1}a^{n-1}$. Congruence $S(n)\equiv-1\pmod n$ means that $n$ divides the integer $S(n)+1$. Giuga's conjecture states \[ S(n)\equiv-1\pmod n\quad\Longleftrightarrow\quad n\text{ is prime}. \] The restriction $n\ge2$ removes the degenerate modulus one. For a prime $n$, each nonzero residue satisfies $a^{n-1}\equiv1\pmod n$, so the forward primality implication is the substantive part: no composite integer should satisfy the displayed sum congruence.

The exponent is tied to the tested integer, and the sum runs over every integer from $1$ to $n-1$, including residues not coprime to $n$. It is therefore not a condition on just the unit group, nor a demand that each summand separately be congruent to one. The asserted criterion is exact, without exceptional composites beyond a cutoff. One can compute the sum using modular exponentiation, but a finite search by itself proves only a bounded range. A disproof requires a composite $n$, together with enough factorization and modular calculation to certify both its compositeness and the sum congruence.

Short English statement

Can a composite integer pass the primality test obtained by summing the (n minus one)st powers of every nonzero residue modulo n?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3360 (OPG-37411), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Giuga's Conjecture on Primality. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/giugas_conjecture_on_primality.