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.

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. https://www.bernoulli.org/~bk/irrpairord.pdf.

Kellner, B. C. "The Equivalence of Giuga's and Agoh's Conjectures." 15 Sep 2004. https://arxiv.org/abs/math.NT/0409259.

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 20-21, 1989.