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Giuga's Conjecture


If n>1 and

 n|1^(n-1)+2^(n-1)+...+(n-1)^(n-1)+1,

is n necessarily a prime? In other words, defining

 s_n=sum_(k=1)^(n-1)k^(n-1),

does there exist a composite n such that s_n=-1 (mod n)? It is known that s_n=-1 (mod n) iff for each prime divisor p of n, (p-1)|(n/p-1) and p|(n/p-1) (Giuga 1950, Borwein et al. 1996); therefore, any counterexample must be squarefree. A composite integer n satisfies s_n=-1 (mod n) iff it is both a Carmichael number and a Giuga number. Giuga showed that there are no exceptions to the conjecture up to 10^(1000). This was later improved to 10^(1700) (Bedocchi 1985) and 10^(13800) (Borwein et al. 1996).

Kellner (2002) provided a short proof of the equivalence of Giuga's and Agoh's conjectures. The combined conjecture can be described by a sum of fractions.


See also

Agoh's Conjecture

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References

Bedocchi, E. "The Z(sqrt(14)) Ring and the Euclidean Algorithm." Manuscripta Math. 53, 199-216, 1985.Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgensohn, R. "Giuga's Conjecture on Primality." Amer. Math. Monthly 103, 40-50, 1996.Giuga, G. "Su una presumibile propertietà caratteristica dei numeri primi." Ist. Lombardo Sci. Lett. Rend. A 83, 511-528, 1950.Kellner, B. C. Über irreguläre Paare höherer Ordnungen. Diplomarbeit. Göttingen, Germany: Mathematischen Institut der Georg August Universität zu Göttingen, 2002. http://www.bernoulli.org/~bk/irrpairord.pdf.Kellner, B. C. "The Equivalence of Giuga's and Agoh's Conjectures." 15 Sep 2004. http://arxiv.org/abs/math.NT/0409259.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 20-21, 1989.

Referenced on Wolfram|Alpha

Giuga's Conjecture

Cite this as:

Weisstein, Eric W. "Giuga's Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GiugasConjecture.html

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