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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.