Goldbach's conjecture - Revision history

Vipul: /* Relation with other facts and conjectures */

2010-02-21T19:08:33Z

Relation with other facts and conjectures

← Older revision		Revision as of 19:08, 21 February 2010
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	\| [[Stronger than::Schnirelmann's theorem on Goldbach's conjecture]] \|\| every even integer greater than <math>2</math> is expressible as the sum of at most <math>300,000</math> primes \|\| proved		\| [[Stronger than::Schnirelmann's theorem on Goldbach's conjecture]] \|\| every even integer greater than <math>2</math> is expressible as the sum of at most <math>300,000</math> primes \|\| proved
	\|-		\|-
	\| [[Stronger than::Chen's theorem]] \|\| every sufficiently large even integer is expressible as the sum of a [[prime number]] and a [[semiprime]], i.e., a number that is either prime or is a product of two primes \|\| proved		\| [[Stronger than::Chen's theorem on Goldbach's conjecture]] \|\| every sufficiently large even integer is expressible as the sum of a [[prime number]] and a [[semiprime]], i.e., a number that is either prime or is a product of two primes \|\| proved
	\|}		\|}

Vipul: /* Relation with other facts and conjectures */

2010-02-21T19:07:47Z

Relation with other facts and conjectures

← Older revision		Revision as of 19:07, 21 February 2010
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	! Name of conjecture/fact !! Statement !! Status		! Name of conjecture/fact !! Statement !! Status
	\|-		\|-
	\| [[Stronger than::Goldbach~~'s weak~~ conjecture]] \|\| every odd integer greater than <math>7</math> is expressible as the sum of three odd primes \|\| corollary of [[generalized Riemann hypothesis]]		\| [[Stronger than::Weak Goldbach conjecture]] \|\| every odd integer greater than <math>7</math> is expressible as the sum of three odd primes \|\| corollary of [[generalized Riemann hypothesis]]
	\|-		\|-
	\| [[Stronger than::Vinogradov's theorem]] \|\| every ''sufficiently large'' odd integer is expressible as the sum of three odd primes. Also finds that there are many such triples. The current bound of sufficiently large is approximately <math>e^{3100} \simeq 2 \cdot 10^{1346}</math> \|\| proved		\| [[Stronger than::Vinogradov's theorem]] \|\| every ''sufficiently large'' odd integer is expressible as the sum of three odd primes. Also finds that there are many such triples. The current bound of sufficiently large is approximately <math>e^{3100} \simeq 2 \cdot 10^{1346}</math> \|\| proved

Vipul: /* Relation with other facts and conjectures */

2010-02-09T01:17:28Z

Relation with other facts and conjectures

← Older revision		Revision as of 01:17, 9 February 2010
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	===Weaker conjectures and facts===		===Weaker conjectures and facts===
	* [[Stronger than::Goldbach's weak conjecture]] ~~is the statement that~~ every odd integer greater than <math>7</math> is expressible as the sum of three odd primes~~. This is currently known to follow from the~~ [[generalized Riemann hypothesis]].
	** [[Stronger than::Vinogradov's theorem]] ~~states that~~ every ''sufficiently large'' odd integer is expressible as the sum of three odd primes. The current bound of sufficiently large is approximately <math>e^{3100} \simeq 2 \cdot 10^{1346}</math>.		{\| class="sortable" border="1"
	** [[~~Chaohua~~'s strengthening of Vinogradov's theorem]] ~~states that~~ we can choose ~~these odd~~ primes to be roughly equal.		! Name of conjecture/fact !! Statement !! Status
	* [[Stronger than::Schnirelmann's theorem on Goldbach's conjecture]]~~: This states that~~ every even integer greater than <math>2</math> is expressible as the sum of at most <math>300,000</math> primes.		\|-
	* [[Stronger than::Chen's theorem]] ~~which states that~~ every sufficiently large even integer is expressible as the sum of a [[prime number]] and a [[semiprime]], i.e., a number that is either prime or is a product of two primes.		\| [[Stronger than::Goldbach's weak conjecture]] \|\| every odd integer greater than <math>7</math> is expressible as the sum of three odd primes \|\| corollary of [[generalized Riemann hypothesis]]
			\|-
			\| [[Stronger than::Vinogradov's theorem]] \|\| every ''sufficiently large'' odd integer is expressible as the sum of three odd primes. Also finds that there are many such triples. The current bound of sufficiently large is approximately <math>e^{3100} \simeq 2 \cdot 10^{1346}</math> \|\| proved
			\|-
			\| [[Haselgrove's strengthening of Vinogradov's theorem]] \|\| in the statement of Vinogradov's theorem, we can choose the three primes to be ''roughly equal'' \|\| proved
			\|-
			\| [[Chaohua's strengthening of Vinogradov's theorem]] \|\| numerically strengthens Haselgrove's statement \|\| proved
			\|-
			\| [[Stronger than::Schnirelmann's theorem on Goldbach's conjecture]] \|\| every even integer greater than <math>2</math> is expressible as the sum of at most <math>300,000</math> primes \|\| proved
			\|-
			\| [[Stronger than::Chen's theorem]] \|\| every sufficiently large even integer is expressible as the sum of a [[prime number]] and a [[semiprime]], i.e., a number that is either prime or is a product of two primes \|\| proved
			\|}

Vipul: /* Relation with other facts and conjectures */

2009-04-18T18:13:11Z

Relation with other facts and conjectures

← Older revision		Revision as of 18:13, 18 April 2009
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	===Weaker conjectures and facts===		===Weaker conjectures and facts===
	* [[Stronger than::Goldbach's weak conjecture]] is the statement that every odd integer greater than <math>5</math> is expressible as the sum of three odd primes. This is currently known to follow from the [[generalized Riemann hypothesis]].		* [[Stronger than::Goldbach's weak conjecture]] is the statement that every odd integer greater than <math>7</math> is expressible as the sum of three odd primes. This is currently known to follow from the [[generalized Riemann hypothesis]].
	** [[Stronger than::Vinogradov's theorem]] states that every ''sufficiently large'' odd integer is expressible as the sum of three odd primes. The current bound of sufficiently large is approximately <math>e^{3100} \simeq 2 \cdot 10^{1346}</math>.		** [[Stronger than::Vinogradov's theorem]] states that every ''sufficiently large'' odd integer is expressible as the sum of three odd primes. The current bound of sufficiently large is approximately <math>e^{3100} \simeq 2 \cdot 10^{1346}</math>.
	** [[Chaohua's strengthening of Vinogradov's theorem]] states that we can choose these odd primes to be roughly equal.		** [[Chaohua's strengthening of Vinogradov's theorem]] states that we can choose these odd primes to be roughly equal.
	* [[Stronger than::Schnirelmann's theorem on Goldbach's conjecture]]: This states that every even integer greater than <math>2</math> is expressible as the sum of at most <math>300,000</math> primes.		* [[Stronger than::Schnirelmann's theorem on Goldbach's conjecture]]: This states that every even integer greater than <math>2</math> is expressible as the sum of at most <math>300,000</math> primes.
	* [[Stronger than::Chen's theorem]] which states that every sufficiently large even integer is expressible as the sum of a [[prime number]] and a [[semiprime]], i.e., a number that is either prime or is a product of two primes.		* [[Stronger than::Chen's theorem]] which states that every sufficiently large even integer is expressible as the sum of a [[prime number]] and a [[semiprime]], i.e., a number that is either prime or is a product of two primes.

Vipul at 22:54, 6 April 2009

2009-04-06T22:54:32Z

← Older revision		Revision as of 22:54, 6 April 2009
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	* Every even integer greater than <math>2</math> is expressible as a sum of two (possibly equal) primes.		* Every even integer greater than <math>2</math> is expressible as a sum of two (possibly equal) primes.
	* Every even integer greater than <math>4</math> is expressible as a sum of two (possibly equal) odd primes.		* Every even integer greater than <math>4</math> is expressible as a sum of two (possibly equal) odd primes.

			A partition of an even integer as a sum of two primes is termed a ''Goldbach partition''.

	==Relation with other facts and conjectures==		==Relation with other facts and conjectures==

Vipul: /* Relation with other facts and conjectures */

2009-04-06T22:49:21Z

Relation with other facts and conjectures

← Older revision		Revision as of 22:49, 6 April 2009
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	==Relation with other facts and conjectures==		==Relation with other facts and conjectures==

			===Weaker conjectures and facts===
	* [[Stronger than::Goldbach's weak conjecture]] is the statement that every odd integer greater than <math>5</math> is expressible as the sum of three odd primes. This is currently known to follow from the [[generalized Riemann hypothesis]].		* [[Stronger than::Goldbach's weak conjecture]] is the statement that every odd integer greater than <math>5</math> is expressible as the sum of three odd primes. This is currently known to follow from the [[generalized Riemann hypothesis]].
	* [[Stronger than::Vinogradov's theorem]] states that every ''sufficiently large'' odd integer is expressible as the sum of three odd primes. The current bound of sufficiently large is approximately <math>e^{3100} \simeq 2 \cdot 10^{1346}</math>.		** [[Stronger than::Vinogradov's theorem]] states that every ''sufficiently large'' odd integer is expressible as the sum of three odd primes. The current bound of sufficiently large is approximately <math>e^{3100} \simeq 2 \cdot 10^{1346}</math>.
	* [[Chaohua's strengthening of Vinogradov's theorem]] states that we can choose these odd primes to be roughly equal.		** [[Chaohua's strengthening of Vinogradov's theorem]] states that we can choose these odd primes to be roughly equal.
	* [[Stronger than::Schnirelmann's theorem on Goldbach's conjecture]]: This states that every even integer greater than <math>2</math> is expressible as the sum of at most <math>300,000</math> primes.		* [[Stronger than::Schnirelmann's theorem on Goldbach's conjecture]]: This states that every even integer greater than <math>2</math> is expressible as the sum of at most <math>300,000</math> primes.
			* [[Stronger than::Chen's theorem]] which states that every sufficiently large even integer is expressible as the sum of a [[prime number]] and a [[semiprime]], i.e., a number that is either prime or is a product of two primes.

Vipul: Created page with '{{additive partition conjecture}} ==Statement== The conjecture has the following equivalent forms: * Every even integer greater than $2$ is expressible as a sum of ...'

2009-04-06T22:47:46Z

Created page with '{{additive partition conjecture}} ==Statement== The conjecture has the following equivalent forms: * Every even integer greater than <math>2</math> is expressible as a sum of ...'

New page

{{additive partition conjecture}}

==Statement==

The conjecture has the following equivalent forms:

* Every even integer greater than <math>2</math> is expressible as a sum of two (possibly equal) primes.
* Every even integer greater than <math>4</math> is expressible as a sum of two (possibly equal) odd primes.

==Relation with other facts and conjectures==

* [[Stronger than::Goldbach's weak conjecture]] is the statement that every odd integer greater than <math>5</math> is expressible as the sum of three odd primes. This is currently known to follow from the [[generalized Riemann hypothesis]].
* [[Stronger than::Vinogradov's theorem]] states that every ''sufficiently large'' odd integer is expressible as the sum of three odd primes. The current bound of sufficiently large is approximately <math>e^{3100} \simeq 2 \cdot 10^{1346}</math>.
* [[Chaohua's strengthening of Vinogradov's theorem]] states that we can choose these odd primes to be roughly equal.
* [[Stronger than::Schnirelmann's theorem on Goldbach's conjecture]]: This states that every even integer greater than <math>2</math> is expressible as the sum of at most <math>300,000</math> primes.

Goldbach's conjecture - Revision history

Vipul: /* Relation with other facts and conjectures */

Vipul: /* Relation with other facts and conjectures */

Vipul: /* Relation with other facts and conjectures */

Vipul: /* Relation with other facts and conjectures */

Vipul at 22:54, 6 April 2009

Vipul: /* Relation with other facts and conjectures */

Vipul: Created page with '{{additive partition conjecture}} ==Statement== The conjecture has the following equivalent forms: * Every even integer greater than 2 is expressible as a sum of ...'

Vipul: Created page with '{{additive partition conjecture}} ==Statement== The conjecture has the following equivalent forms: * Every even integer greater than $2$ is expressible as a sum of ...'