Any Composite Number with for all Prime Divisors of . is a Giuga
number Iff

where is the Totient Function and Iff

is a Giuga number Iff

where is a Bernoulli Number and is the Totient Function. Every counterexample to Giuga's conjecture is a contradiction to Argoh's Conjecture and vice versa. The smallest known Giuga numbers are 30 (3 factors), 858, 1722 (4 factors), 66198 (5 factors), 2214408306, 24423128562 (6 factors), 432749205173838, 14737133470010574, 550843391309130318 (7 factors),

244197000982499715087866346, 554079914617070801288578559178(8 factors), ... (Sloane's A007850).

It is not known if there are an infinite number of Giuga numbers. All the above numbers have sum minus product equal to 1, and any Giuga number of higher order must have at least 59 factors. The smallest Odd Giuga number must have at least nine Prime factors.

**References**

Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgensohn, R. ``Giuga's Conjecture on Primality.''
*Amer. Math. Monthly* **103**, 40-50, 1996.

Sloane, N. J. A. Sequence A007850 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

© 1996-9

1999-05-25