Properties
The prime factors of a Giuga number must be distinct. If divides, then it follows that, where is divisible by . Hence, would not be divisible by, and thus would not be a Giuga number.
Thus, only square-free integers can be Giuga numbers. For example, the factors of 60 are 2, 2, 3 and 5, and 60/2 - 1 = 29, which is not divisible by 2. Thus, 60 is not a Giuga number.
This rules out squares of primes, but semiprimes cannot be Giuga numbers either. For if, with primes, then, so will not divide, and thus is not a Giuga number.
| Are there infinitely many Giuga numbers? |
All known Giuga numbers are even. If an odd Giuga number exists, it must be the product of at least 14 primes. It is not known if there are infinitely many Giuga numbers.
It has been conjectured by Paolo P. Lava (2009) that Giuga numbers are the solutions of the differential equation n'=n+1, being n' the arithmetic derivative of n.
Read more about this topic: Giuga Number
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