Innumber theory, aGiuga number is acomposite number${\displaystyle n}$ such that for each of its distinctprime factors${\displaystyle p_i}$ we have${\displaystyle p_i|\left(\frac{n}{p_i}-1\right)}$, or equivalently such that for each of its distinctprime factorsp_i we have${\displaystyle p_i^2|(n-p_i)}$.

The Giuga numbers are named after the mathematicianGiuseppe Giuga, and relate tohis conjecture on primality.

Alternative definition for aGiuga number due toTakashi Agoh is: acomposite numbern is aGiuga numberif and only if the congruence

${\displaystyle n{B}_{\phi (n)}\equiv -1(\mathrm{mod}n)}$

holds true, whereB is aBernoulli number and${\displaystyle \phi (n)}$ isEuler's totient function.

An equivalent formulation due toGiuseppe Giuga is: acomposite numbern is aGiuga number if and only if the congruence

${\displaystyle \sum _{i=1}^{n-1}{i}^{\phi (n)}\equiv -1(\mathrm{mod}n)}$

and if and only if

${\displaystyle \sum _{p|n}\frac{1}{p}-\prod _{p|n}\frac{1}{p}\in \mathbb{N}.}$

All known Giuga numbersn in fact satisfy the stronger condition

${\displaystyle \sum _{p|n}\frac{1}{p}-\prod _{p|n}\frac{1}{p}=1.}$

The sequence of Giuga numbers begins

30, 858, 1722, 66198, 2214408306, 24423128562, 432749205173838, … (sequenceA007850 in theOEIS).

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${\displaystyle p^2}$ divides${\displaystyle n}$, then it follows that${\displaystyle \frac{n}{p}-1=m-1}$, where${\displaystyle m=n/p}$ is divisible by${\displaystyle p}$. Hence,${\displaystyle m-1}$ would not be divisible by${\displaystyle p}$, and thus${\displaystyle n}$ would not be a Giuga number.

Thus, onlysquare-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, butsemiprimes cannot be Giuga numbers either. For if${\displaystyle n=p_1{p}_2}$, with primes, then, so${\displaystyle p_2}$ will not divide${\displaystyle \frac{n}{p_2}-1}$, and thus${\displaystyle n}$ 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 14primes. It is not known if there are infinitely many Giuga numbers.

It has been conjectured by Paolo P. Lava (2009) that Giuga numbers are the solutions of the differential equationn' = n+1, wheren' is thearithmetic derivative ofn. (For square-free numbers${\displaystyle n=\prod _i{p}_i}$,${\displaystyle n^{\prime }=\sum _i\frac{n}{p_i}}$, son' = n+1 is just the last equation in the above sectionDefinitions, multiplied byn.)

José Mª Grau and Antonio Oller-Marcén have shown that an integern is a Giuga number if and only if it satisfiesn' = a n + 1 for some integera > 0, wheren' is thearithmetic derivative ofn. (Again,n' = a n + 1 is identical to the third equation inDefinitions, multiplied byn.)

• Carmichael number
• Primary pseudoperfect number
• Znám's problem

• Weisstein, Eric W."Giuga Number".MathWorld.
• Borwein, D.;Borwein, J. M.;Borwein, P. B.; Girgensohn, R. (1996)."Giuga's Conjecture on Primality"(PDF).American Mathematical Monthly.103 (1):40–50.CiteSeerX 10.1.1.586.1424.doi:10.2307/2975213.JSTOR 2975213.Zbl 0860.11003. Archived fromthe original(PDF) on 2005-05-31.
• Balzarotti, Giorgio; Lava, Paolo P. (2010).Centotre curiosità matematiche. Milan: Hoepli Editore. p. 129.ISBN 978-88-203-4556-3.

• Integer sequences
• Unsolved problems in number theory

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