Linnik’s theorem on the least prime in an arithmetic progression
E747889
Linnik’s theorem on the least prime in an arithmetic progression is a result in analytic number theory that gives an explicit upper bound, depending only on the modulus, for the size of the smallest prime in any given coprime residue class.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Linnik’s theorem on the least prime in an arithmetic progression canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8644725 — 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: Linnik’s theorem on the least prime in an arithmetic progression Context triple: [Deuring–Heilbronn phenomenon, relatedTo, Linnik’s theorem on the least prime in an arithmetic progression]
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A.
Dirichlet's theorem on arithmetic progressions
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.
-
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.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
-
D.
Bombieri–Vinogradov theorem
The Bombieri–Vinogradov theorem is a major result in analytic number theory that gives strong average estimates for the distribution of prime numbers in arithmetic progressions, approaching what is predicted by the Generalized Riemann Hypothesis.
-
E.
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.
- 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: Linnik’s theorem on the least prime in an arithmetic progression Target entity description: Linnik’s theorem on the least prime in an arithmetic progression is a result in analytic number theory that gives an explicit upper bound, depending only on the modulus, for the size of the smallest prime in any given coprime residue class.
-
A.
Dirichlet's theorem on arithmetic progressions
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.
-
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.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
-
D.
Bombieri–Vinogradov theorem
The Bombieri–Vinogradov theorem is a major result in analytic number theory that gives strong average estimates for the distribution of prime numbers in arithmetic progressions, approaching what is predicted by the Generalized Riemann Hypothesis.
-
E.
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.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
result in number theory
ⓘ
theorem in analytic number theory ⓘ |
| assertsExistenceOf | absolute constant L > 0 independent of q and a ⓘ |
| concerns |
distribution of primes in arithmetic progressions
ⓘ
least prime in an arithmetic progression ⓘ upper bounds for least primes in residue classes ⓘ |
| conclusionInvolves | least prime p ≡ a (mod q) ⓘ |
| field |
analytic number theory
ⓘ
number theory ⓘ |
| gives | explicit upper bound for the least prime in a coprime residue class modulo q ⓘ |
| hasConsequence |
effective version of Dirichlet’s theorem with explicit dependence on q
ⓘ
existence of primes in short intervals within arithmetic progressions up to C q^L ⓘ |
| hasParameter |
Linnik exponent L
ⓘ
modulus q ⓘ residue class a modulo q ⓘ |
| hasRefinementsBy |
Heath-Brown
ⓘ
linked to:
Roger Heath-Brown
Xylouris ⓘ other analytic number theorists ⓘ |
| implies | every coprime residue class modulo q contains a prime not exceeding C q^L ⓘ |
| improvesOn | trivial exponential bounds for least primes in arithmetic progressions ⓘ |
| involves | Linnik exponent ⓘ |
| isDiscussedIn |
monographs on analytic number theory
ⓘ
research articles on least primes in arithmetic progressions ⓘ |
| isRelatedTo |
Chebotarev density theorem
ⓘ
Dirichlet’s theorem on arithmetic progressions ⓘ Linnik’s dispersion method ⓘ least prime in a given residue class problem ⓘ zero-free regions of Dirichlet L-functions ⓘ |
| isUsedIn |
applications to computational number theory involving prime search in residue classes
ⓘ
studies of primes in arithmetic progressions with large moduli ⓘ |
| isWeakerThan | Generalized Riemann Hypothesis bounds for least primes in arithmetic progressions ⓘ |
| namedAfter |
Yuri Vladimirovich Linnik
ⓘ
linked to:
Yury Linnik
|
| originallyProvedBy |
Yuri Linnik
ⓘ
linked to:
Yury Linnik
|
| originalProofUsed | Linnik’s dispersion method ⓘ |
| quantifiesOver |
all integers a with gcd(a,q)=1
ⓘ
all integers q ≥ 1 ⓘ |
| requiresCondition | residue class a is coprime to modulus q ⓘ |
| statesThat | there exists a constant L such that the least prime p ≡ a (mod q) with (a,q)=1 satisfies p ≤ C q^L for some constant C ⓘ |
| strengthens | Dirichlet’s theorem by giving an explicit upper bound for the least prime ⓘ |
| subfield | multiplicative number theory ⓘ |
| typeOfBound | polynomial bound in q ⓘ |
| usesMethod |
analytic techniques involving L-functions
ⓘ
zero-density estimates for Dirichlet L-functions ⓘ zero-free regions for Dirichlet L-functions ⓘ |
| yearProved | 1944 ⓘ |
How these facts were elicited
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Input
Subject: Linnik’s theorem on the least prime in an arithmetic progression Description of subject: Linnik’s theorem on the least prime in an arithmetic progression is a result in analytic number theory that gives an explicit upper bound, depending only on the modulus, for the size of the smallest prime in any given coprime residue class.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
Deuring–Heilbronn phenomenon
→
relatedTo
→
Linnik’s theorem on the least prime in an arithmetic progression
ⓘ