Giuga number

A Giuga number is a composite number n such that for each of its distinct prime factors p_i we have [math]\displaystyle{ p_i | \left({n \over p_i} - 1\right) }[/math], or equivalently such that for each of its distinct prime factors p_i we have [math]\displaystyle{ p_i^2 | (n - p_i) }[/math]. 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

[math]\displaystyle{ nB_{\varphi(n)} \equiv -1 \pmod n }[/math]

holds true, where B is a Bernoulli number and [math]\displaystyle{ \varphi(n) }[/math] is Euler's totient function.

An equivalent formulation due to Giuseppe Giuga is: a composite number n is a Giuga number if and only if the congruence

[math]\displaystyle{ \sum_{i=1}^{n-1} i^{\varphi(n)} \equiv -1 \pmod n }[/math]

and if and only if

[math]\displaystyle{ \sum_{p|n} \frac{1}{p} - \prod_{p|n} \frac{1}{p} \in \mathbb{N}. }[/math]

All known Giuga numbers n in fact satisfy the stronger condition

[math]\displaystyle{ \sum_{p|n} \frac{1}{p} - \prod_{p|n} \frac{1}{p} = 1. }[/math]

Examples

The sequence of Giuga numbers begins

30, 858, 1722, 66198, 2214408306, 24423128562, 432749205173838, … (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

• 30/2 - 1 = 14, which is divisible by 2,
• 30/3 - 1 = 9, which is 3 squared, and
• 30/5 - 1 = 5, the third prime factor itself.

Properties

The prime factors of a Giuga number must be distinct. If [math]\displaystyle{ p^2 }[/math] divides [math]\displaystyle{ n }[/math], then it follows that [math]\displaystyle{ {n \over p} - 1 = m-1 }[/math], where [math]\displaystyle{ m=n/p }[/math] is divisible by [math]\displaystyle{ p }[/math]. Hence, [math]\displaystyle{ m-1 }[/math] would not be divisible by [math]\displaystyle{ p }[/math], and thus [math]\displaystyle{ n }[/math] 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 [math]\displaystyle{ n=p_1p_2 }[/math], with [math]\displaystyle{ p_1\lt p_2 }[/math] primes, then [math]\displaystyle{ {n \over p_2} - 1 =p_1 - 1 \lt p_2 }[/math], so [math]\displaystyle{ p_2 }[/math] will not divide [math]\displaystyle{ {n \over p_2} - 1 }[/math], and thus [math]\displaystyle{ n }[/math] is not a Giuga number.

Unsolved problem in mathematics: Are there infinitely many Giuga numbers? (more unsolved problems in mathematics)

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 [math]\displaystyle{ n = \prod_i {p_i} }[/math], [math]\displaystyle{ n' = \sum_i \frac{n}{p_i} }[/math], so n' = n+1 is just the last equation in the above section Definitions, multiplied by n.)

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.)

See also

• Carmichael number
• Primary pseudoperfect number
• Znám's problem

References

• Weisstein, Eric W.. "Giuga Number". http://mathworld.wolfram.com/GiugaNumber.html. 
• Borwein, D.; Borwein, J. M.; Borwein, P. B.; Girgensohn, R. (1996). "Giuga's Conjecture on Primality". American Mathematical Monthly 103 (1): 40–50. doi:10.2307/2975213. http://www.math.uwo.ca/~dborwein/cv/giuga.pdf. 
• Balzarotti, Giorgio; Lava, Paolo P. (2010). Centotre curiosità matematiche. Milan: Hoepli Editore. p. 129. ISBN 978-88-203-4556-3.