Dirichlet's theorem on arithmetic progressions

E466245

result in analytic number theory theorem in number theory

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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (4)

Label	Occurrences
Dirichlet’s theorem on arithmetic progressions	7
Dirichlet's theorem on arithmetic progressions canonical	6
Dirichlet’s theorem on primes in arithmetic progressions	3
Dirichlet’s theorem by giving an explicit upper bound for the least prime	1

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]

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
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 ⓘ
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 ⓘ linked to: Peter Gustav Lejeune Dirichlet
provedBy	Johann Peter Dirichlet ⓘ linked to: Peter Gustav Lejeune Dirichlet
relatedConcept	Euler product formula ⓘ linked to: Euler product formula for the Riemann zeta function distribution of prime numbers ⓘ non-vanishing of L-functions on the line Re(s)=1 ⓘ reduced residue system modulo n ⓘ
relatedTo	Chebotarev density theorem ⓘ Dirichlet L-function ⓘ linked to: Dirichlet L-functions Dirichlet character ⓘ linked to: Dirichlet characters 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 ⓘ Dirichlet characters ⓘ Euler product ⓘ linked to: Euler product formula for the Riemann zeta function L-series ⓘ characters modulo n ⓘ complex analysis ⓘ non-vanishing of L-functions at s = 1 ⓘ
yearProved	1837 ⓘ

How these facts were elicited

Referenced by (17)

Full triples — surface form annotated when it differs from this entity's canonical label.

Peter Gustav Lejeune Dirichlet → notableWork → Dirichlet's theorem on arithmetic progressions ⓘ

Fermat's theorem on sums of two squares → relatedTo → Dirichlet's theorem on arithmetic progressions ⓘ

Multiplicative Number Theory → studiesProperty → Dirichlet’s theorem on primes in arithmetic progressions ⓘ