Editing Agoh–Giuga conjecture

{{Short description|Number theory conjecture}}
In [[number theory]] the '''Agoh–Giuga conjecture''' on the [[Bernoulli number]]s ''B''<sub>''k''</sub> postulates that ''p'' is a [[prime number]] [[if and only if]] 

:<math>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>1^{p-1}+2^{p-1}+ \cdots +(p-1)^{p-1} \equiv -1 \pmod p</math>

which may also be written as

:<math>\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>a^{p-1} \equiv 1 \pmod p</math>

for <math>a = 1,2,\dots,p-1</math>, and the equivalence follows, since <math>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 number|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, in a work of 2001 showed that a possible counterexample should be a number ''n'' greater than&nbsp;&nbsp;10<sup>36067</sup> 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>(p-1)! \equiv -1 \pmod p,</math>

which may also be written as

:<math>\prod_{i=1}^{p-1} i \equiv -1 \pmod p.</math>

For an odd prime p we have

:<math>\prod_{i=1}^{p-1} i^{p-1} \equiv (-1)^{p-1} \equiv 1 \pmod p,</math>

and for p=2 we have

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

and

:<math>\prod_{i=1}^{p-1} i^{p-1} \equiv 1 \pmod p.</math>

== See also ==

* {{slink|Bernoulli number#Arithmetical properties of the Bernoulli numbers}}

==References==
{{reflist}}
* {{cite journal | last=Giuga | first=Giuseppe | title=Su una presumibile proprietà caratteristica dei numeri primi | language=Italian | journal=Ist.Lombardo Sci. Lett., Rend., Cl. Sci. Mat. Natur. | volume=83 | pages=511–518 | year=1951 | issn=0375-9164 | zbl=0045.01801 }}
* {{cite journal | last=Agoh | first=Takashi | title=On Giuga's conjecture | journal=[[Manuscripta Mathematica]] | volume=87 | number=4 | pages=501–510 | year=1995 | zbl=0845.11004 | doi=10.1007/bf02570490}}
* {{cite journal | author1-link=David Borwein | last1=Borwein | first1=D. | author2-link=Jonathan Borwein | last2=Borwein | first2=J. M. | author3-link=Peter Borwein | last3=Borwein | first3=P. B. | last4=Girgensohn | first4=R. | title=Giuga's Conjecture on Primality | journal=[[American Mathematical Monthly]] | volume=103 | issue=1 | pages=40–50 | year=1996 | zbl=0860.11003 | url=http://www.math.uwo.ca/~dborwein/cv/giuga.pdf | doi=10.2307/2975213 | access-date=2005-05-29 | archive-url=https://web.archive.org/web/20050531164907/http://www.math.uwo.ca/~dborwein/cv/giuga.pdf | archive-date=2005-05-31 | url-status=dead | jstor=2975213 | citeseerx=10.1.1.586.1424 }}
* {{cite journal | last=Sorini | first=Laerte | title=Un Metodo Euristico per la Soluzione della Congettura di Giuga | language=Italian | journal=Quaderni di Economia, Matematica e Statistica, DESP, Università di Urbino Carlo Bo | volume=68 | year=2001 | issn=1720-9668 }}

{{Prime number conjectures}}

{{DEFAULTSORT:Agoh-Giuga conjecture}}
[[Category:Conjectures about prime numbers]]
[[Category:Unsolved problems in number theory]]