Goldbach's conjecture: Difference between revisions

Revision as of 01:17, 9 February 2010

Statement

The conjecture has the following equivalent forms:

Every even integer greater than $2$ is expressible as a sum of two (possibly equal) primes.
Every even integer greater than $4$ 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

Weaker conjectures and facts

Name of conjecture/fact	Statement	Status
Goldbach's weak conjecture	every odd integer greater than $7$ is expressible as the sum of three odd primes	corollary of generalized Riemann hypothesis
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 $e^{3100}\simeq 2\cdot 10^{1346}$	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
Schnirelmann's theorem on Goldbach's conjecture	every even integer greater than $2$ is expressible as the sum of at most $300,000$ primes	proved
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

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 ===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
+|}