Giuga Number/Examples/30

Example of Giuga Number

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

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

Proof

We have that:

$30 = 2 \times 3 \times 5$

Then:

\(\ds \dfrac 1 2 + \dfrac 1 3 + \dfrac 1 5\)

\(=\)

\(\ds \frac {3 \times 5 + 2 \times 5 + 2 \times 3} {2 \times 3 \times 5}\)

\(\ds \)

\(=\)

\(\ds \frac {15 + 10 + 6} {30}\)

\(\ds \)

\(=\)

\(\ds \frac {31} {30}\)

\(\ds \)

\(=\)

\(\ds 1 + \frac 1 {30}\)

Hence the result by definition of Giuga number.

$\blacksquare$

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): $30$