Giuga's Conjecture

If n>1 and


is n necessarily a prime? In other words, defining


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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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., B. C. "The Equivalence of Giuga's and Agoh's Conjectures." 15 Sep 2004., 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.

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