Vipul: Created page with "==Statement== ===Formulation in terms of Bernoulli numbers=== This formulation dates to Agoh in 1990. It states that a natural number $p$ is a [[prime number]..."

2012-06-23T00:17:15Z

Created page with "==Statement== ===Formulation in terms of Bernoulli numbers=== This formulation dates to Agoh in 1990. It states that a natural number <math>p</math> is a [[prime number]..."

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==Statement==

===Formulation in terms of Bernoulli numbers===

This formulation dates to Agoh in 1990. It states that a [[natural number]] <math>p</math> is a [[prime number]] if and only if we have:

<math>pB_{p-1} \equiv -1 \pmod p</math>

where <math>B_{p-1}</math> is the [[Bernoulli number]] corresponding to <math>p - 1</math>.

===Formulation in terms of power sums===

This formulation dates to Giuga in the 1950s.

A [[natural number]] <math>p</math> is a [[prime number]] if and only if we have:

<math>\sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p</math>

Note that one direction is immediate: if <math>p</math> is a [[prime number]], then the above congruence holds true as a consequence of [[Fermat's little theorem]]. The other direction is conjectural and open.

===Formulation in terms of Giuga numbers===

There is no (composite) natural number that is both a [[Carmichael number]] and a [[Giuga number]]. Note that this is equivalent to the power sums formulation because composite number satisfies the power sums condition iff it is both a Carmichael number and a Giuga number.

Agoh-Giuga conjecture - Revision history

Vipul: Created page with "==Statement== ===Formulation in terms of Bernoulli numbers=== This formulation dates to Agoh in 1990. It states that a natural number p is a [[prime number]..."

Vipul: Created page with "==Statement== ===Formulation in terms of Bernoulli numbers=== This formulation dates to Agoh in 1990. It states that a natural number $p$ is a [[prime number]..."