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# Giuga's Conjecture

If and

is necessarily a prime? In other words, defining

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

Agoh's Conjecture

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## References

Bedocchi, E. "The 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