Giuga Number
Any Composite Number 
with 
for all Prime Divisors 
of . is a Giuganumber 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 conjectureis 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 (7factors), 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, andany Giuga number of higher order must have at least 59 factors. The smallest Odd Giuga number must have at least ninePrime factors.
See also Argoh's Conjecture, Bernoulli Number, Totient Function
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 A007850in ``The On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.
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