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 [ - 1]. Another test is if the congruence [nB_ \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 first few Giuga numbers are

30, 858, 1722, 66198, 2214408306, ... (sequence 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 need to be distinct. If we allow [n = p^2], it follows that [ - 1 = p - 1], which will clearly not divide p. Thus, no squareful number can be a Giuga number. For example, the factors of 60 are 2, 2, 3 and 5, and 60/2 - 1 = 29, which is obviously not divisible by 2. Suffice it to say that the other prime factors of 60 also fail the test.

This effectively rules out squares of primes, but semiprimes p₁p₂ can not be Giuga numbers either. Even if the smaller factor p₁ passes the test (and it certainly will if it happens to be 2), the larger factor p₂ will fail the test, since [ - 1 < p_2].

All known Giuga numbers are even. If an odd Giuga number exists, it must be the product of at least nine primes.

References

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

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