Goldbach's conjecture: Difference between revisions
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! Name of conjecture/fact !! Statement !! Status | ! Name of conjecture/fact !! Statement !! Status | ||
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| [[Stronger than::Goldbach | | [[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]] | ||
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| [[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 | ||
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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 | ||
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| [[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 | ||
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Latest revision as of 19:08, 21 February 2010
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
| Name of conjecture/fact | Statement | Status |
|---|---|---|
| Weak Goldbach conjecture | every odd integer greater than 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 | 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 is expressible as the sum of at most primes | proved |
| 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 |
