Dirichlet's approximation theorem - Revision history

en>Nxavar: ce

2014-12-15T18:06:33Z

← Older revision		Revision as of 20:06, 15 December 2014
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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>.		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.<br><br>Also visit my homepage [http://chatmast.com/index.php?do=/BookerSipessk/info/ at home std testing]

	~~The Giuga numbers are named after the mathematician [[Giuseppe Giuga]],~~ and ~~relate to [[Giuga's conjecture\|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~~

	~~:<math>nB_{\varphi(n)} \equiv -1 \pmod n</math>~~

	~~holds true, where ''B'' is a [[Bernoulli number]] and <math>\varphi(n)</math> 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~~

	~~:<math>\sum_{i=1}^{n-1} i^{\varphi(n)} \equiv -1 \pmod n</math>~~

	~~and if and only if~~

	~~:<math>\sum_{p\|n} \frac{1}{p} - \prod_{p\|n} \frac{1}{p} \in \mathbb{N}.</math>~~

	~~All known Giuga numbers ''n'' in fact satisfy the stronger condition~~

	~~:<math>\sum_{p\|n} \frac{1}{p} - \prod_{p\|n} \frac{1}{p} = 1~~.~~</math>~~

	~~== Examples ==~~
	~~The sequence of Giuga numbers begins~~

	~~:30, 858, 1722, 66198, 2214408306, … {{OEIS\|id=A007850}}.~~

	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.

	~~== Properties ==~~
	~~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.~~

	~~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~~.

	~~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~~
	~~<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.~~

	~~{{unsolved\|mathematics\|Are there infinitely many Giuga numbers?\|Is there a composite Giuga number that is also a Carmichael number?}}~~
	~~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.~~

	~~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.~~

	~~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.~~

	~~==See also==~~
	*[[Carmichael number]]
	*[[Primary pseudoperfect number]]

	~~==References==~~
	* {{MathWorld\|title=Giuga Number\|urlname=GiugaNumber}}
	* {{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 }}~~
	* {{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 }}

	~~{{Classes of natural numbers}}~~
	~~[[Category:Integer sequences]]~~
	~~[[Category:Unsolved problems in mathematics]~~]

en>Slawekb: Fixed mysterious error.

2013-09-26T11:45:30Z

Fixed mysterious error.

← Older revision		Revision as of 13:45, 26 September 2013
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	~~Greetings~~. The ~~author~~'s ~~name~~ is ~~Phebe~~ and ~~she feels comfortable when individuals use~~ the ~~complete name~~. ~~To collect coins~~ is ~~one~~ of the ~~issues I love most~~. ~~Hiring~~ is ~~my profession~~. ~~California~~ is ~~our birth location~~.<br><br>~~Here~~ is ~~my homepage: home std test kit~~ - [~~https~~://~~hcbsales~~.~~com~~/~~node~~/~~3739 mouse click the following web site~~],		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>.

			The Giuga numbers are named after the mathematician [[Giuseppe Giuga]], and relate to [[Giuga's conjecture\|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

			:<math>nB_{\varphi(n)} \equiv -1 \pmod n</math>

			holds true, where ''B'' is a [[Bernoulli number]] and <math>\varphi(n)</math> 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

			:<math>\sum_{i=1}^{n-1} i^{\varphi(n)} \equiv -1 \pmod n</math>

			and if and only if

			:<math>\sum_{p\|n} \frac{1}{p} - \prod_{p\|n} \frac{1}{p} \in \mathbb{N}.</math>

			All known Giuga numbers ''n'' in fact satisfy the stronger condition

			:<math>\sum_{p\|n} \frac{1}{p} - \prod_{p\|n} \frac{1}{p} = 1.</math>

			== Examples ==
			The sequence of Giuga numbers begins

			:30, 858, 1722, 66198, 2214408306, … {{OEIS\|id=A007850}}.

			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.

			== Properties ==
			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.

			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.

			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
			<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.

			{{unsolved\|mathematics\|Are there infinitely many Giuga numbers?\|Is there a composite Giuga number that is also a Carmichael number?}}
			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.

			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.

			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.

			==See also==
			*[[Carmichael number]]
			*[[Primary pseudoperfect number]]

			==References==
			* {{MathWorld\|title=Giuga Number\|urlname=GiugaNumber}}
			* {{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 }}
			* {{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 }}

			{{Classes of natural numbers}}
			[[Category:Integer sequences]]
			[[Category:Unsolved problems in mathematics]]

en>Thue Siegel Roth: /* See also */ Hurwitz's theorem (number theory)

2012-02-08T14:29:05Z

See also: Hurwitz's theorem (number theory)

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