Dirichlet's theorem on arithmetic progressions
E466245
Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.
All labels observed (3)
How this entity was disambiguated
This entity first appeared as the object of triple T4746233 — 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: Dirichlet's theorem on arithmetic progressions Context triple: [Peter Gustav Lejeune Dirichlet, notableWork, Dirichlet's theorem on arithmetic progressions]
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A.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
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B.
Vinogradov's three-primes theorem
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.
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C.
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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D.
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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E.
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.
- 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: Dirichlet's theorem on arithmetic progressions Target entity description: Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.
-
A.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
-
B.
Vinogradov's three-primes theorem
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.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
result in analytic number theory
ⓘ
theorem in number theory ⓘ |
| appliesTo |
any modulus d ≥ 1
ⓘ
any residue class a modulo d with gcd(a,d)=1 ⓘ |
| assumption | gcd(a,d)=1 for the arithmetic progression a+nd ⓘ |
| classification | deep theorem of classical number theory ⓘ |
| condition | The first term and common difference of the arithmetic progression must be coprime ⓘ |
| consequence |
Distribution of primes among reduced residue classes modulo d is infinite in each class
ⓘ
No congruence class coprime to the modulus can contain only finitely many primes ⓘ |
| doesNotApplyTo | arithmetic progressions where gcd(a,d) > 1 ⓘ |
| domain | arithmetic progressions of integers ⓘ |
| ensures | For any modulus d and residue class a coprime to d, there are infinitely many primes p with p ≡ a (mod d) ⓘ |
| field |
analytic number theory
ⓘ
number theory ⓘ |
| generalizes | Euclid's theorem on the infinitude of primes NERFINISHED ⓘ |
| historicalSignificance |
Introduced Dirichlet characters and L-series into number theory
ⓘ
One of the first major uses of analytic methods in number theory ⓘ |
| implies | There are infinitely many primes congruent to a modulo d for any a coprime to d ⓘ |
| importance | fundamental theorem in multiplicative number theory ⓘ |
| methodOfProof |
analytic methods
ⓘ
use of L-series and complex analysis ⓘ |
| namedAfter | Johann Peter Dirichlet NERFINISHED ⓘ |
| provedBy | Johann Peter Dirichlet NERFINISHED ⓘ |
| relatedConcept |
Euler product formula
NERFINISHED
ⓘ
distribution of prime numbers ⓘ non-vanishing of L-functions on the line Re(s)=1 ⓘ reduced residue system modulo n ⓘ |
| relatedTo |
Chebotarev density theorem
NERFINISHED
ⓘ
Dirichlet L-function NERFINISHED ⓘ Dirichlet character NERFINISHED ⓘ prime number theorem for arithmetic progressions ⓘ |
| specialCase |
Infinitude of primes in the progression -1 mod n for any n with gcd(-1,n)=1
ⓘ
Infinitude of primes in the progression 1 mod 4 ⓘ Infinitude of primes in the progression 1 mod n for any n ⓘ Infinitude of primes in the progression 3 mod 4 ⓘ |
| statement | Every arithmetic progression a, a+d, a+2d, ... with gcd(a,d)=1 contains infinitely many prime numbers ⓘ |
| strengthenedBy | prime number theorem for arithmetic progressions ⓘ |
| typeOfResult | infinitude of primes result ⓘ |
| usesConcept |
Dirichlet L-functions
NERFINISHED
ⓘ
Dirichlet characters NERFINISHED ⓘ Euler product NERFINISHED ⓘ L-series NERFINISHED ⓘ characters modulo n ⓘ complex analysis ⓘ non-vanishing of L-functions at s = 1 ⓘ |
| yearProved | 1837 ⓘ |
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Input
Subject: Dirichlet's theorem on arithmetic progressions Description of subject: Dirichlet's theorem on arithmetic progressions is a fundamental result in number theory stating that any arithmetic progression with first term and difference coprime contains infinitely many prime numbers.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
Fermat's theorem on sums of two squares
→
relatedTo
→
Dirichlet's theorem on arithmetic progressions
ⓘ
this entity surface form:
Dirichlet’s theorem on primes in arithmetic progressions
this entity surface form:
Dirichlet’s theorem on arithmetic progressions
this entity surface form:
Dirichlet’s theorem on arithmetic progressions
this entity surface form:
Dirichlet’s theorem on arithmetic progressions