Legendre’s conjecture on primes between consecutive squares
E695820
Legendre’s conjecture on primes between consecutive squares is an unproven statement in number theory asserting that there is always at least one prime number between any two consecutive perfect squares.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Legendre’s conjecture on primes between consecutive squares canonical | 2 |
| Legendre’s conjecture | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7861125 — 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: Legendre’s conjecture on primes between consecutive squares Context triple: [Adrien-Marie Legendre, knownFor, Legendre’s conjecture on primes between consecutive squares]
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A.
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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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.
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C.
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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D.
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.
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E.
Pólya’s conjecture
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.
- 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: Legendre’s conjecture on primes between consecutive squares Target entity description: Legendre’s conjecture on primes between consecutive squares is an unproven statement in number theory asserting that there is always at least one prime number between any two consecutive perfect squares.
-
A.
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.
-
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.
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.
-
D.
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.
-
E.
Pólya’s conjecture
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
unproven statement in number theory ⓘ |
| assumes | infinitely many squares and potential primes between them ⓘ |
| category |
conjectures about prime numbers
ⓘ
unsolved problems in number theory ⓘ |
| concerns |
distribution of prime numbers
ⓘ
gaps between prime numbers ⓘ |
| countryOfOrigin | France ⓘ |
| currentEvidence | verified for many values of n by computation ⓘ |
| difficulty | considered very hard ⓘ |
| doesNotHave |
known counterexample
ⓘ
known general proof ⓘ |
| equivalentFormulation | For every integer n ≥ 1 there is a prime between n^2 and (n+1)^2 ⓘ |
| field | number theory ⓘ |
| hasForm | quantified statement over all positive integers n ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| impliedBy | Riemann hypothesis together with sufficiently strong zero-density estimates ⓘ |
| implies |
existence of primes in every interval (n^2,(n+1)^2)
ⓘ
there are infinitely many prime numbers ⓘ |
| involvesConcept |
asymptotic density of primes
ⓘ
perfect squares ⓘ prime gaps ⓘ prime numbers ⓘ |
| logicalType | universal-existential statement ⓘ |
| mainStatement | For every positive integer n there exists at least one prime p with n^2 < p < (n+1)^2 ⓘ |
| motivatedBy | observed regularity in tables of primes and squares ⓘ |
| namedAfter | Adrien-Marie Legendre ⓘ |
| openQuestion | whether there can be a prime gap covering an entire interval between consecutive squares ⓘ |
| proposedBy | Adrien-Marie Legendre ⓘ |
| publicationContext | work on the distribution of prime numbers by Adrien-Marie Legendre ⓘ |
| relatedTo |
Bertrand’s postulate
ⓘ
linked to:
Bertrand's postulate
Cramér’s conjecture ⓘ Legendre’s conjecture on the prime-counting function π(x) ⓘ Legendre’s constant in the approximation of π(x) ⓘ Legendre’s empirical formula for π(x) = x/(log x − A) ⓘ Legendre’s work "Essai sur la théorie des nombres" ⓘ
linked to:
Essai sur la théorie des nombres
Riemann hypothesis ⓘ prime number theorem ⓘ problems on primes in polynomial sequences ⓘ |
| status |
open problem
ⓘ
unproven ⓘ |
| subfield |
analytic number theory
ⓘ
prime number theory ⓘ |
| symbolicForm | ∀n ∈ ℕ, ∃p prime such that n^2 < p < (n+1)^2 ⓘ |
| typeOf | conjecture about primes in short intervals ⓘ |
| upperBoundIntervalLength | 2n+1 for interval (n^2,(n+1)^2) ⓘ |
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: Legendre’s conjecture on primes between consecutive squares Description of subject: Legendre’s conjecture on primes between consecutive squares is an unproven statement in number theory asserting that there is always at least one prime number between any two consecutive perfect squares.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.