Agoh–Giuga conjecture

The Agoh–Giuga conjecture states that an integer n greater than one is prime exactly when 1^(n−1) + 2^(n−1) + ⋯ + (n−1)^(n−1) is congruent to −1 modulo n.

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Fermat's little theorem: For prime p and integer a not divisible by p, a^(p−1) is congruent to 1 modulo p. Applying it term by term makes the conjectured congruence immediate for prime n.
Giuseppe Giuga: An Italian mathematician known for a conjecture characterizing prime numbers through congruences and Bernoulli numbers. His proposed primality criterion supplies the Giuga side of the conjecture's name.
Composite number: An integer greater than one that is not prime. A single composite satisfying the congruence would disprove the conjecture.
Primality test: An algorithm or mathematical procedure that determines whether an integer is prime. The conjecture would give a direct, if computationally costly, primality criterion.
Modular arithmetic: Arithmetic in which integers are identified when they have the same remainder upon division by a fixed modulus. The conjecture's defining condition is a congruence modulo n.
Takashi Agoh: A Japanese mathematician who proposed a Bernoulli-number criterion equivalent to Giuga's primality conjecture. His formulation gives the Agoh side of the combined conjecture.
Agoh–Giuga number: A composite integer n for which the sum of the (n−1)st powers of 1 through n−1 is congruent to −1 modulo n. These are precisely the potential counterexamples whose existence would refute the conjecture.
Trial division: A primality-testing method that checks divisibility by integers up to the candidate's square root. It offers a practical elementary alternative to evaluating the conjecture's enormous power sum.
Power sum: A sum of terms obtained by raising a sequence of numbers to a fixed exponent. The conjecture tests primality using the power sum of the integers below n.
Bernoulli numbers: A sequence of rational numbers that appears in power-sum formulas and many areas of number theory. Agoh's equivalent formulation expresses the primality test using the Bernoulli number B_(n−1).