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| A '''Giuga number''' is a [[composite number]] ''n'' such that for each of its distinct [[prime factor]]s ''p''<sub>''i''</sub> we have <math>p_i | ({n \over p_i} - 1)</math>, or equivalently such that for each of its distinct [[prime factor]]s ''p''<sub>''i''</sub> we have <math>p_i^2 | (n - p_i)</math>.
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| The Giuga numbers are named after the mathematician [[Giuseppe Giuga]], and relate to [[Giuga's conjecture|his conjecture]] on primality.
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| == Definitions ==
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| 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
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| :<math>nB_{\varphi(n)} \equiv -1 \pmod n</math>
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| holds true, where ''B'' is a [[Bernoulli number]] and <math>\varphi(n)</math> is [[Euler's totient function]].
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| An equivalent formulation due to [[Giuseppe Giuga]] is: a [[composite number]] ''n'' is a '''Giuga number''' if and only if the congruence
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| :<math>\sum_{i=1}^{n-1} i^{\varphi(n)} \equiv -1 \pmod n</math>
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| and if and only if
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| :<math>\sum_{p|n} \frac{1}{p} - \prod_{p|n} \frac{1}{p} \in \mathbb{N}.</math>
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| All known Giuga numbers ''n'' in fact satisfy the stronger condition
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| :<math>\sum_{p|n} \frac{1}{p} - \prod_{p|n} \frac{1}{p} = 1.</math>
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| == Examples ==
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| The sequence of Giuga numbers begins
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| :30, 858, 1722, 66198, 2214408306, … {{OEIS|id=A007850}}.
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| For example, 30 is a Giuga number since its prime factors are 2, 3 and 5, and we can verify that | |
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| * 30/2 - 1 = 14, which is divisible by 2,
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| * 30/3 - 1 = 9, which is 3 squared, and
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| * 30/5 - 1 = 5, the third prime factor itself.
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| == Properties ==
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| The prime factors of a Giuga number must be distinct. If <math>p^2</math> divides <math>n</math>, then it follows that <math>{n \over p} - 1 = n'-1</math>, where <math>n'</math> is divisible by <math>p</math>. Hence, <math>n'-1</math> would not be divisible by <math>p</math>, and thus <math>n</math> would not be a Giuga number.
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| Thus, only [[square-free integer]]s 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.
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| This rules out squares of primes, but [[semiprime]]s cannot be Giuga numbers either. For if <math>n=p_1p_2</math>, with <math>p_1<p_2</math> primes, then
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| <math>{n \over p_2} - 1 =p_1 - 1 <p_2</math>, so <math>p_2</math> will not divide <math>{n \over p_2} - 1 </math>, and thus <math>n</math> is not a Giuga number.
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| {{unsolved|mathematics|Are there infinitely many Giuga numbers?|Is there a composite Giuga number that is also a Carmichael number?}}
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| All known Giuga numbers are even. If an odd Giuga number exists, it must be the product of at least 14 [[prime number|prime]]s. It is not known if there are infinitely many Giuga numbers.
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| It has been conjectured by Paolo P. Lava (2009) that Giuga numbers are the solutions of the differential equation n'=n+1, where n' is the [[arithmetic derivative]] of n.
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| José Mª Grau and Antonio Oller-Marcén have shown that an integer n is a Giuga number if and only if it satisfies n'= an +1 for some integer a>0, where n' is the [[arithmetic derivative]] of n.
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| ==See also==
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| *[[Carmichael number]]
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| *[[Primary pseudoperfect number]]
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| ==References==
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| * {{MathWorld|title=Giuga Number|urlname=GiugaNumber}}
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| * {{cite journal | author1-link=David Borwein | last1=Borwein | first1=D. | author2-link=Jonathan Borwein | last2=Borwein | first2=J. M. | author3-link=Peter Borwein | last3=Borwein | first3=P. B. | last4=Girgensohn | first4=R. | title=Giuga's Conjecture on Primality | journal=[[American Mathematical Monthly]] | volume=103 | pages=40–50 | year=1996 | zbl=0860.11003 | url=http://www.math.uwo.ca/~dborwein/cv/giuga.pdf }}
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| * {{cite book | first1=Giorgio | last1=Balzarotti | first2=Paolo P. | last2=Lava | title=Centotre curiosità matematiche | location=Milan | publisher=Hoepli Editore | year=2010 | isbn=978-88-203-4556-3 | page=129 }}
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| {{Classes of natural numbers}}
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| [[Category:Integer sequences]]
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| [[Category:Unsolved problems in mathematics]]
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Greetings! I am Marvella and I feel comfortable when individuals use the complete name. Playing baseball is the pastime he will by no means quit performing. For many years he's been operating as a receptionist. Minnesota has always been his house but his wife desires them to move.
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