Giuga number

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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, ... (sequence A007850 in 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.

The prime factors of a Giuga number must be distinct. If $p 2$ divides $n$ , then it follows that ${n \over p} - 1 = n'-1$ , where $n'$ is divisible by $p$ . Hence, $n' - 1$ 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 $n = p 1 p 2$ , with $p 1 < p 2$ primes, then ${n \over p_2} - 1 =p_1 - 1 <p_2$ , so $p 2$ will not divide ${n \over p_2} - 1$ , and thus $n$ is not a Giuga number.

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.