Pólya’s conjecture
E637315
Pólya’s conjecture is a disproven hypothesis in number theory that proposed a specific long-term sign pattern for the summatory Möbius function, suggesting it would eventually remain nonpositive.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Pólya conjecture (disproved) | 1 |
| Pólya’s conjecture canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7030868 — 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: Pólya’s conjecture Context triple: [George Pólya, notableIdea, Pólya’s conjecture]
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A.
Erdős–Turán conjecture
The Erdős–Turán conjecture is an unsolved problem in additive number theory asserting that any subset of the positive integers with divergent sum of reciprocals must contain arbitrarily long arithmetic progressions.
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B.
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.
-
C.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
-
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.
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.
- 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: Pólya’s conjecture Target entity description: Pólya’s conjecture is a disproven hypothesis in number theory that proposed a specific long-term sign pattern for the summatory Möbius function, suggesting it would eventually remain nonpositive.
-
A.
Erdős–Turán conjecture
The Erdős–Turán conjecture is an unsolved problem in additive number theory asserting that any subset of the positive integers with divergent sum of reciprocals must contain arbitrarily long arithmetic progressions.
-
B.
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.
-
C.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
-
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.
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.
- F. None of above. chosen
Statements (39)
| Predicate | Object |
|---|---|
| instanceOf |
disproven conjecture
ⓘ
mathematical conjecture ⓘ number theory conjecture ⓘ |
| assumes | that M(x) does not become positive infinitely often ⓘ |
| contradictedBy | explicit large positive values of M(x) ⓘ |
| disproofMethod | existence of counterexamples for large x ⓘ |
| disproofYear | 1958 ⓘ |
| disprovedBy | C. Brian Haselgrove ⓘ |
| equivalentlyStates | the summatory Möbius function is eventually nonpositive ⓘ |
| field | number theory ⓘ |
| hasCounterexamples | values of x for which M(x) > 0 ⓘ |
| hasInfluenceOn |
analytic number theory
ⓘ
study of sign changes of M(x) ⓘ subsequent research on the Möbius function ⓘ |
| hasMathematicsSubjectClassification |
11M26
ⓘ
11N37 ⓘ |
| implies | a strong restriction on the sign pattern of M(x) ⓘ |
| involvesConcept |
Riemann hypothesis
ⓘ
Riemann zeta function ⓘ distribution of prime numbers ⓘ sign changes ⓘ |
| involvesFunction |
Möbius function
ⓘ
summatory Möbius function M(x) ⓘ |
| isAbout |
asymptotic behavior of M(x)
ⓘ
long-term sign pattern of the summatory Möbius function ⓘ |
| isWeakerThan | Mertens conjecture ⓘ |
| languageOfOriginalPublication | German ⓘ |
| mainSubject |
Mertens function
ⓘ
summatory Möbius function ⓘ |
| motivatedBy | heuristics from the Riemann hypothesis ⓘ |
| namedAfter | George Pólya ⓘ |
| proposedBy | George Pólya ⓘ |
| publicationYear | 1919 ⓘ |
| relatedConjecture | Mertens conjecture ⓘ |
| relatedTo |
oscillation of arithmetic functions
ⓘ
partial sums of multiplicative functions ⓘ |
| statedIn | paper by George Pólya on the zeros of the Riemann zeta function and related functions ⓘ |
| statementForm | M(x) ≤ 0 for all sufficiently large x ⓘ |
| status | disproved ⓘ |
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
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Pólya’s conjecture Description of subject: Pólya’s conjecture is a disproven hypothesis in number theory that proposed a specific long-term sign pattern for the summatory Möbius function, suggesting it would eventually remain nonpositive.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Pólya’s conjecture