Giuga number

A Giuga number is a composite number "n" such that each of its distinct prime factors "p"_"i" is a divisor of $\{n\; over\; p\_i\}\; -\; 1$. Another test is if the congruence $nB\_\{phi(n)\}\; equiv\; -1\; pmod\; n$ holds true, where "B" is a Bernoulli number. The Giuga numbers are named after the mathematician Giuseppe Giuga, and relate to his conjecture on primality.

The sequence of Giuga numbers begins

:30, 858, 1722, 66198, 2214408306, … OEIS|id=A007850.

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.

The prime factors of a Giuga number must be distinct. If $p^2$ divides $n$, then it follows that $\{n\; over\; p\}\; -\; 1\; =\; n\text{'}-1$, where $n\text{'}$ is divisible by $p$. Hence, $n\text{'}-1$ would not be divisible by $p$, and thus $n$ 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 $n=p\_1p\_2$, with

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.

ee also

*Carmichael number

References

* Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgensohn, R. "Giuga's Conjecture on Primality." "American Mathematical Monthly" 103, 40-50, 1996.