Goldbach's conjecture: Difference between revisions
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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> | * [[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. | ||
Revision as of 18:13, 18 April 2009
Template:Additive partition conjecture
Statement
The conjecture has the following equivalent forms:
- Every even integer greater than is expressible as a sum of two (possibly equal) primes.
- Every even integer greater than 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
- Goldbach's weak conjecture is the statement that every odd integer greater than is expressible as the sum of three odd primes. This is currently known to follow from the generalized Riemann hypothesis.
- 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 .
- Chaohua's strengthening of Vinogradov's theorem states that we can choose these odd primes to be roughly equal.
- Schnirelmann's theorem on Goldbach's conjecture: This states that every even integer greater than is expressible as the sum of at most primes.
- 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.
