Agoh–Giuga conjecture

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

:$1^\{p-1\}+2^\{p-1\}+\; \backslash cdots\; +(p-1)^\{p-1\}\; \backslash equiv\; -1\; \backslash pmod\; p$

which may also be written as

:$\backslash sum\_\{i=1\}^\{p-1\}\; i^\{p-1\}\; \backslash equiv\; -1\; \backslash pmod\; p.$

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

:$a^\{p-1\}\; \backslash equiv\; 1\; \backslash pmod\; p$

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

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, 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.

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

:$(p-1)!\; \backslash equiv\; -1\; \backslash pmod\; p,$

which may also be written as

:$\backslash prod\_\{i=1\}^\{p-1\}\; i\; \backslash equiv\; -1\; \backslash pmod\; p.$

For an odd prime p we have

:$\backslash prod\_\{i=1\}^\{p-1\}\; i^\{p-1\}\; \backslash equiv\; (-1)^\{p-1\}\; \backslash equiv\; 1\; \backslash pmod\; p,$

and for p=2 we have

:$\backslash prod\_\{i=1\}^\{p-1\}\; i^\{p-1\}\; \backslash equiv\; (-1)^\{p-1\}\; \backslash equiv\; 1\; \backslash pmod\; p.$

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

:$\backslash sum\_\{i=1\}^\{p-1\}\; i^\{p-1\}\; \backslash equiv\; -1\; \backslash pmod\; p$

and

:$\backslash prod\_\{i=1\}^\{p-1\}\; i^\{p-1\}\; \backslash equiv\; 1\; \backslash pmod\; p.$

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

{{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