Vinogradov's three-primes theorem
E451525
Vinogradov's three-primes theorem is a landmark result in analytic number theory proving that every sufficiently large odd integer can be expressed as the sum of three prime numbers.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
result in analytic number theory
ⓘ
theorem ⓘ |
| appearsIn | Vinogradov's book "The Method of Trigonometrical Sums in the Theory of Numbers" NERFINISHED ⓘ |
| assumption | no unproven hypotheses such as the Riemann hypothesis ⓘ |
| category |
additive prime number theory
ⓘ
theorems about prime numbers ⓘ |
| concerns |
odd integers
ⓘ
prime numbers ⓘ representations of integers as sums of primes ⓘ |
| conclusion | existence of three primes whose sum equals the given odd integer ⓘ |
| field |
analytic number theory
ⓘ
number theory ⓘ |
| hasApproximateForm | Asymptotic formula for the number of representations of a large odd integer as a sum of three primes NERFINISHED ⓘ |
| hasConsequence | every sufficiently large odd integer is the sum of three odd primes ⓘ |
| historicalImportance | first major unconditional result towards the ternary Goldbach conjecture ⓘ |
| implies | weak form of the odd Goldbach conjecture for sufficiently large integers ⓘ |
| inspired | subsequent research on additive problems involving primes ⓘ |
| involves |
asymptotic analysis
ⓘ
sieve methods (indirectly in later refinements) ⓘ |
| isLandmarkIn | 20th-century analytic number theory ⓘ |
| methodUsed |
Hardy–Littlewood circle method
NERFINISHED
ⓘ
estimates for trigonometric sums ⓘ exponential sums over primes ⓘ |
| namedAfter | Ivan Matveyevich Vinogradov NERFINISHED ⓘ |
| originalLanguage | Russian ⓘ |
| provedBy | Ivan Matveyevich Vinogradov NERFINISHED ⓘ |
| publishedIn | Doklady Akademii Nauk SSSR NERFINISHED ⓘ |
| quantifier | sufficiently large odd integer ⓘ |
| refinedBy | work of Ramaré and others on explicit bounds ⓘ |
| relatedResult | Helfgott's proof of the full ternary Goldbach conjecture ⓘ |
| relatedTo |
Goldbach conjecture
NERFINISHED
ⓘ
ternary Goldbach problem NERFINISHED ⓘ |
| statement | Every sufficiently large odd integer can be expressed as the sum of three prime numbers. ⓘ |
| strengthenedBy |
later work removing or lowering the bound on "sufficiently large"
ⓘ
results of Chen Jingrun on Goldbach-type problems NERFINISHED ⓘ |
| subfield | additive number theory ⓘ |
| topicOf | many monographs on analytic number theory ⓘ |
| typeOf | ternary Goldbach theorem NERFINISHED ⓘ |
| usesConcept |
Dirichlet characters
NERFINISHED
ⓘ
L-functions NERFINISHED ⓘ Weyl differencing NERFINISHED ⓘ major arcs and minor arcs decomposition ⓘ zero-free regions for L-functions ⓘ |
| usesTool |
estimates for exponential sums over primes
ⓘ
orthogonality of characters ⓘ |
| yearProved | 1937 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.