In number theory, a Giuga number is a composite number n {\displaystyle n} such that for each of its distinct prime factors p i {\displaystyle p_{i}} we have p i | ( n p i − 1 ) {\displaystyle p_{i}|\left({n \over p_{i}}-1\right)} , or equivalently such that for each of its distinct prime factors pi we have p i 2 | ( n − p i ) {\displaystyle p_{i}^{2}|(n-p_{i})} . For example, 30 = 2 × 3 × 5 is a Giuga number since we can verify that:
30/2 − 1 = 14 = 2 × 7, 30/3 − 1 = 9 = 32, and 30/5 − 1 = 5. The Giuga numbers are named after the mathematician Giuseppe Giuga, and relate to his conjecture on primality.
Definitions Alternative definition for a Giuga number due to Takashi Agoh is: a composite number n is a Giuga number if and only if the congruence
n B φ ( n ) ≡ − 1 ( mod n ) {\displaystyle nB_{\varphi (n)}\equiv -1{\pmod {n}}}
holds true, where B is a Bernoulli number and φ ( n ) {\displaystyle \varphi (n)} is Euler's totient function. An equivalent formulation due to Giuseppe Giuga is: a composite number n is a Giuga number if and only if the congruence
∑ i = 1 n − 1 i φ ( n ) ≡ − 1 ( mod n ) {\displaystyle \sum _{i=1}^{n-1}i^{\varphi (n)}\equiv -1{\pmod {n}}}
and if and only if
∑ p | n 1 p − ∏ p | n 1 p ∈ N . {\displaystyle \sum _{p|n}{\frac {1}{p}}-\prod _{p|n}{\frac {1}{p}}\in \mathbb {N} .}
All known Giuga numbers n in fact satisfy the stronger condition
∑ p | n 1 p − ∏ p | n 1 p = 1. {\displaystyle \sum _{p|n}{\frac {1}{p}}-\prod _{p|n}{\frac {1}{p}}=1.}
List of numbers Thirteen Giuga numbers are known. The list is complete up to the 12th term and for numbers with 8 or fewer prime factors, but it is unknown if there is a Giuga number between the 12th and 13th terms.
Properties The prime factors of a Giuga number must be distinct. If p 2 {\displaystyle p^{2}} divides n {\displaystyle n} , then it follows that n p − 1 = m − 1 {\displaystyle {n \over p}-1=m-1} , where m = n / p {\displaystyle m=n/p} is divisible by p {\displaystyle p} . Hence, m − 1 {\displaystyle m-1} would not be divisible by p {\displaystyle p} , and thus n {\displaystyle 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 {\displaystyle n=p_{1}p_{2}} , with p 1 < p 2 {\displaystyle p_{1}<p_{2}} primes, then
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