Agoh–Giuga conjecture

In number theory the Agoh–Giuga conjecture on the Bernoulli numbers B_k postulates that p is a prime number if and only if

pB_{p-1} \equiv -1 \pmod p.

It is named after Takashi Agoh and Giuseppe Giuga.

Equivalent formulation

The conjecture as stated above is due to Takashi Agoh (1990); an equivalent formulation is due to Giuseppe Giuga, from 1950, to the effect that p is prime if

1^{p-1}+2^{p-1}+ \cdots +(p-1)^{p-1} \equiv -1 \pmod p

which may also be written as

\sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p.

It is trivial to show that p being prime is sufficient for the second equivalence to hold, since if p is prime, Fermat's little theorem states that

a^{p-1} \equiv 1 \pmod p

for $a = 1,2,\dots,p-1$ , and the equivalence follows, since $p-1 \equiv -1 \pmod p.$

Status

The statement is still a conjecture since it has not yet been proven that if a number n is not prime (that is, n is composite), then the formula does not hold. It has been shown that a composite number n satisfies the formula if and only if it is both a Carmichael number and a Giuga number, and that if such a number exists, it has at least 13,800 digits (Borwein, Borwein, Borwein, Girgensohn 1996).

Relation to Wilson's theorem

The Agoh–Giuga conjecture bears a similarity to Wilson's theorem, which has been proven to be true. Wilson's theorem states that a number p is prime if and only if

(p-1)! \equiv -1 \pmod p,

which may also be written as

\prod_{i=1}^{p-1} i \equiv -1 \pmod p.

For an odd prime p we have

\prod_{i=1}^{p-1} i^{p-1} \equiv (-1)^{p-1} \equiv 1 \pmod p,

and for p=2 we have

\prod_{i=1}^{p-1} i^{p-1} \equiv (-1)^{p-1} \equiv 1 \pmod p.

So, the truth of the Agoh–Giuga conjecture combined with Wilson's theorem would give: a number p is prime if and only if

\sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p

and

\prod_{i=1}^{p-1} i^{p-1} \equiv 1 \pmod p.

References

Agoh, Takashi (1995). "On Giuga's conjecture". Manuscripta Mathematica 87 (4): 501–510. doi:10.1007/bf02570490. Zbl 0845.11004.
Borwein, D.; Borwein, J. M.; Borwein, P. B.; Girgensohn, R. (1996). "Giuga's Conjecture on Primality" (PDF). American Mathematical Monthly 103: 40–50. doi:10.2307/2975213. Zbl 0860.11003.
Giuga, Giuseppe (1951). "Su una presumibile proprietà caratteristica dei numeri primi". Ist.Lombardo Sci. Lett., Rend., Cl. Sci. Mat. Natur. (in Italian) 83: 511–518. ISSN 0375-9164. Zbl 0045.01801.

Prime number conjectures

Hardy–Littlewood 1st 2nd Agoh–Giuga Andrica's Artin's Bateman–Horn Brocard's Bunyakovsky Chinese hypothesis Cramér's Dickson's Elliott–Halberstam Firoozbakht's Gilbreath's Grimm's Landau's problems Goldbach's weak Legendre's Twin prime Legendre's constant Lemoine's Mersenne Oppermann's Polignac's Pólya Redmond–Sun Schinzel's hypothesis H Waring's prime number

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