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

[math]\displaystyle{ pB_{p-1} \equiv -1 \pmod p. }[/math]

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 and only if

[math]\displaystyle{ 1^{p-1}+2^{p-1}+ \cdots +(p-1)^{p-1} \equiv -1 \pmod p }[/math]

which may also be written as

[math]\displaystyle{ \sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p. }[/math]

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

[math]\displaystyle{ a^{p-1} \equiv 1 \pmod p }[/math]

for [math]\displaystyle{ a = 1,2,\dots,p-1 }[/math], and the equivalence follows, since [math]\displaystyle{ p-1 \equiv -1 \pmod p. }[/math]

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). Laerte Sorini, finally, in a work of 2001 showed that a possible counterexample should be a number n greater than  10³⁶⁰⁶⁷ which represents the limit suggested by Bedocchi for the demonstration technique specified by Giuga to his own conjecture.

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

[math]\displaystyle{ (p-1)! \equiv -1 \pmod p, }[/math]

which may also be written as

[math]\displaystyle{ \prod_{i=1}^{p-1} i \equiv -1 \pmod p. }[/math]

For an odd prime p we have

[math]\displaystyle{ \prod_{i=1}^{p-1} i^{p-1} \equiv (-1)^{p-1} \equiv 1 \pmod p, }[/math]

and for p=2 we have

[math]\displaystyle{ \prod_{i=1}^{p-1} i^{p-1} \equiv (-1)^{p-1} \equiv 1 \pmod p. }[/math]

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

[math]\displaystyle{ \sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p }[/math]

and

[math]\displaystyle{ \prod_{i=1}^{p-1} i^{p-1} \equiv 1 \pmod p. }[/math]

References

• Giuga, Giuseppe (1951). "Su una presumibile proprietà caratteristica dei numeri primi" (in Italian). Ist.Lombardo Sci. Lett., Rend., Cl. Sci. Mat. Natur. 83: 511–518. ISSN 0375-9164. 
• Agoh, Takashi (1995). "On Giuga's conjecture". Manuscripta Mathematica 87 (4): 501–510. doi:10.1007/bf02570490. 
• Borwein, D.; Borwein, J. M.; Borwein, P. B.; Girgensohn, R. (1996). "Giuga's Conjecture on Primality". American Mathematical Monthly 103 (1): 40–50. doi:10.2307/2975213. http://www.math.uwo.ca/~dborwein/cv/giuga.pdf. Retrieved 2005-05-29. 
• Sorini, Laerte (2001). "Un Metodo Euristico per la Soluzione della Congettura di Giuga" (in Italian). Quaderni di Economia, Matematica e Statistica, DESP, Università di Urbino Carlo Bo 68. ISSN 1720-9668.