A Giuga number is a composite number n such that for each of its distinct prime factors pi we have
Contents
The Giuga numbers are named after the mathematician Giuseppe Giuga, and relate to his conjecture on primality.
Definitions
Alternative definition for a Giuga number due to Takashi Agoh is: a composite number n is a Giuga number if and only if the congruence
holds true, where B is a Bernoulli number and
An equivalent formulation due to Giuseppe Giuga is: a composite number n is a Giuga number if and only if the congruence
and if and only if
All known Giuga numbers n in fact satisfy the stronger condition
Examples
The sequence of Giuga numbers begins
30, 858, 1722, 66198, 2214408306, … (sequence A007850 in the OEIS).For example, 30 is a Giuga number since its prime factors are 2, 3 and 5, and we can verify that
Properties
The prime factors of a Giuga number must be distinct. If
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
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, where n' is the arithmetic derivative of n. (For square-free numbers
José Mª Grau and Antonio Oller-Marcén have shown that an integer n is a Giuga number if and only if it satisfies n' = a n + 1 for some integer a > 0, where n' is the arithmetic derivative of n. (Again, n' = a n + 1 is identical to the third equation in Definitions, multiplied by n.)