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.
All labels observed (4)
How this entity was disambiguated
This entity first appeared as the object of triple T4552393 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Vinogradov's three-primes theorem Context triple: [Hardy–Littlewood circle method, notableApplication, Vinogradov's three-primes theorem]
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A.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
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B.
Über die Anzahl der Primzahlen unter einer gegebenen Grösse
Über die Anzahl der Primzahlen unter einer gegebenen Grösse is Bernhard Riemann’s seminal 1859 paper that introduced the Riemann zeta function and laid the foundations of analytic number theory, including the famous Riemann Hypothesis.
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C.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
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D.
Green–Tao theorem
The Green–Tao theorem is a landmark result in number theory proving that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
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E.
Three Pearls of Number Theory
Three Pearls of Number Theory is a classic mathematical text that presents three elegant and accessible problems in number theory, illustrating deep ideas through simple, beautifully explained examples.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Vinogradov's three-primes theorem Target entity description: 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.
-
A.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
-
B.
Über die Anzahl der Primzahlen unter einer gegebenen Grösse
Über die Anzahl der Primzahlen unter einer gegebenen Grösse is Bernhard Riemann’s seminal 1859 paper that introduced the Riemann zeta function and laid the foundations of analytic number theory, including the famous Riemann Hypothesis.
-
C.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
-
D.
Green–Tao theorem
The Green–Tao theorem is a landmark result in number theory proving that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
-
E.
Three Pearls of Number Theory
Three Pearls of Number Theory is a classic mathematical text that presents three elegant and accessible problems in number theory, illustrating deep ideas through simple, beautifully explained examples.
- F. None of above. chosen
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" ⓘ |
| 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
ⓘ
linked to:
Vinogradov's three-primes theorem
|
| 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
ⓘ
estimates for trigonometric sums ⓘ exponential sums over primes ⓘ |
| namedAfter | Ivan Matveyevich Vinogradov ⓘ |
| originalLanguage | Russian ⓘ |
| provedBy | Ivan Matveyevich Vinogradov ⓘ |
| publishedIn | Doklady Akademii Nauk SSSR ⓘ |
| 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
ⓘ
ternary Goldbach problem ⓘ
linked to:
Vinogradov's three-primes theorem
|
| 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 ⓘ |
| subfield | additive number theory ⓘ |
| topicOf | many monographs on analytic number theory ⓘ |
| typeOf |
ternary Goldbach theorem
ⓘ
linked to:
weak Goldbach conjecture
|
| usesConcept |
Dirichlet characters
ⓘ
L-functions ⓘ Weyl differencing ⓘ major arcs and minor arcs decomposition ⓘ zero-free regions for L-functions ⓘ |
| usesTool |
estimates for exponential sums over primes
ⓘ
orthogonality of characters ⓘ |
| yearProved | 1937 ⓘ |
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Subject: Vinogradov's three-primes theorem Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Vinogradov's three-primes theorem
linked to: Vinogradov's three-primes theorem
Vinogradov's three-primes theorem
→
hasApproximateForm
→
Asymptotic formula for the number of representations of a large odd integer as a sum of three primes
ⓘ
linked to: Vinogradov's three-primes theorem