Bateman–Horn conjecture
E451528
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bateman–Horn conjecture canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4552478 — 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: Bateman–Horn conjecture Context triple: [Hardy–Littlewood conjectures, relatedTo, Bateman–Horn conjecture]
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A.
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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B.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
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C.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
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D.
Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
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E.
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.
- 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: Bateman–Horn conjecture Target entity description: 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.
-
A.
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.
-
B.
Ramanujan–Petersson conjecture
The Ramanujan–Petersson conjecture is a fundamental statement in number theory and the theory of modular forms that predicts strong bounds on the Fourier coefficients of modular cusp forms, with deep connections to automorphic forms and the Langlands program.
-
C.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
-
D.
Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
unproven statement in number theory ⓘ |
| appliesTo | finite sets of polynomials ⓘ |
| assumes |
Generalized Riemann Hypothesis does not directly imply it
ⓘ
no congruence obstruction to primality for the polynomial values ⓘ |
| conclusionType | asymptotic formula ⓘ |
| describes | asymptotic density of integers for which given polynomials take prime values ⓘ |
| domain | polynomials with integer coefficients ⓘ |
| field | number theory ⓘ |
| generalizes |
Bunyakovsky conjecture
NERFINISHED
ⓘ
Dirichlet’s theorem on arithmetic progressions NERFINISHED ⓘ Hardy–Littlewood conjecture F NERFINISHED ⓘ Hardy–Littlewood prime k-tuple conjecture NERFINISHED ⓘ prime number theorem for arithmetic progressions ⓘ |
| gives | constant depending on the given polynomials ⓘ |
| hasAbbreviation | BH conjecture NERFINISHED ⓘ |
| hasConsequence |
predictions for record prime-producing polynomials
ⓘ
quantitative estimates for gaps between prime values of polynomials ⓘ |
| implies |
infinitely many Sophie Germain primes
ⓘ
infinitely many prime k-tuples for any admissible pattern ⓘ infinitely many primes of the form n^2+1 under suitable conditions ⓘ infinitely many primes represented by any admissible irreducible polynomial ⓘ infinitely many twin primes ⓘ |
| involves |
Euler product over primes
ⓘ
local density factors at each prime ⓘ product of correction factors reflecting congruence obstructions ⓘ |
| isPartOf | conjectural framework for distribution of primes in polynomial sequences ⓘ |
| languageOfOriginalPublication | English ⓘ |
| mathematicsSubjectClassification |
11N05
ⓘ
11N32 ⓘ |
| motivatedBy | observed distribution of prime values of polynomials ⓘ |
| namedAfter |
Paul T. Bateman
NERFINISHED
ⓘ
Roger A. Horn NERFINISHED ⓘ |
| namedEntityType | mathematical object ⓘ |
| predicts | frequency of simultaneous prime values of several polynomials ⓘ |
| publication | paper by Bateman and Horn in Transactions of the American Mathematical Society ⓘ |
| relatedTo | Schinzel’s hypothesis H NERFINISHED ⓘ |
| requiresCondition |
polynomials are irreducible over the integers
ⓘ
polynomials do not share a fixed common factor for all integer inputs ⓘ polynomials have positive leading coefficient ⓘ |
| status | unproven ⓘ |
| strongerThan | Schinzel’s hypothesis H in many formulations NERFINISHED ⓘ |
| subfield |
analytic number theory
ⓘ
prime number theory ⓘ |
| usedFor |
estimating counts of primes in polynomial sequences
ⓘ
heuristic predictions about prime-generating polynomials ⓘ |
| yearProposed | 1962 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Bateman–Horn conjecture Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.