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.
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 NERFINISHED ⓘ |
| openQuestion | whether there can be a prime gap covering an entire interval between consecutive squares ⓘ |
| proposedBy | Adrien-Marie Legendre NERFINISHED ⓘ |
| publicationContext | work on the distribution of prime numbers by Adrien-Marie Legendre ⓘ |
| relatedTo |
Bertrand’s postulate
NERFINISHED
ⓘ
Cramér’s conjecture NERFINISHED ⓘ 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" NERFINISHED ⓘ Riemann hypothesis NERFINISHED ⓘ prime number theorem NERFINISHED ⓘ 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) ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.