Definition:Giuga Number

Definition

Let $n$ be a positive integer.

$n$ is a Giuga number if and only if:

$\ds \sum_{p \mathop \divides n} \frac 1 p - \prod_{p \mathop \divides n} \frac 1 p \in \Z_{\ge 0}$

where:

$p$ denotes a prime number

$p \divides n$ therefore denotes that $p$ is a divisor of $n$.

Sequence

The sequence of Giuga numbers begins:

$30, 858, 1722, 66 \, 198, 2 \, 214 \, 408 \, 306, 24 \, 423 \, 128 \, 562, 432 \, 749 \, 205 \, 173 \, 838, \ldots$

Examples

30

$30$ is a Giuga number (the smallest):

$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 {30} = 1$

858

$858$ is a Giuga number:

$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 {11} + \dfrac 1 {13} - \dfrac 1 {858} = 1$

1722

$1722$ is a Giuga number:

$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 7 + \dfrac 1 {41} - \dfrac 1 {1722} = 1$

$244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 346$

$244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 346$ is a Giuga number:

$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 {11} + \dfrac 1 {23} + \dfrac 1 {31} + \dfrac 1 {47 \, 137} + \dfrac 1 {28 \, 282 \, 147} + \dfrac 1 {3 \, 892 \, 535 \, 183} - \dfrac 1 {244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 346} = 1$

Source of Name

This entry was named for Giuseppe Giuga.

Sources

• Jan. 1996: D. Borwein, J.M. Borwein, P.B. Borwein and R. Girgensohn: Giuga's Conjecture on Primality (Amer. Math. Monthly Vol. 103, no. 1: pp. 40 – 50)  www.jstor.org/stable/2975213

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $30$

• Weisstein, Eric W. "Giuga Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GiugaNumber.html