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- Legendre's_conjecture abstract "Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n2 and (n + 1)2 for every positive integer n. The conjecture is one of Landau's problems (1912) and remains unsolved.The prime number theorem suggests the actual number of primes between n2 and (n + 1)2 OEIS A014085 is about n/ln(n), i.e. about as many as the number of primes less than or equal to n.If Legendre's conjecture is true, the gap between any two successive primes would be . Two stronger conjectures, Andrica's conjecture and Oppermann's conjecture, also both imply that the gaps have the same magnitude. Harald Cramér conjectured that the gap is always much smaller, . If Cramér's conjecture is true, Legendre's conjecture would follow for all sufficiently large numbers. Cramér also proved that the Riemann hypothesis implies a weaker bound of on the size of the largest prime gaps. Legendre's conjecture implies that at least one prime can be found in every half revolution of the Ulam spiral.Baker, Harman and Pintz proved that there is a prime in the interval for all large . This effectively proves the conjecture up to 1012 by solving .Because the conjecture follows from Andrica's conjecture, it suffices to check that each prime gap starting at p is smaller than A table of maximal prime gaps shows that the conjecture holds to 1018. A counterexample near 1018 would require a prime gap fifty million times the size of the average gap.".
- Legendre's_conjecture thumbnail Plot_of_number_of_primes_between_consecutive_squares.png?width=300.
- Legendre's_conjecture wikiPageID "2620450".
- Legendre's_conjecture wikiPageRevisionID "603264332".
- Legendre's_conjecture hasPhotoCollection Legendre's_conjecture.
- Legendre's_conjecture title "Legendre's conjecture".
- Legendre's_conjecture urlname "LegendresConjecture".
- Legendre's_conjecture subject Category:Conjectures_about_prime_numbers.
- Legendre's_conjecture subject Category:Unsolved_problems_in_mathematics.
- Legendre's_conjecture type Abstraction100002137.
- Legendre's_conjecture type Attribute100024264.
- Legendre's_conjecture type Cognition100023271.
- Legendre's_conjecture type Concept105835747.
- Legendre's_conjecture type Condition113920835.
- Legendre's_conjecture type ConjecturesAboutPrimeNumbers.
- Legendre's_conjecture type Content105809192.
- Legendre's_conjecture type Difficulty114408086.
- Legendre's_conjecture type Hypothesis105888929.
- Legendre's_conjecture type Idea105833840.
- Legendre's_conjecture type Problem114410605.
- Legendre's_conjecture type PsychologicalFeature100023100.
- Legendre's_conjecture type Speculation105891783.
- Legendre's_conjecture type State100024720.
- Legendre's_conjecture type UnsolvedProblemsInMathematics.
- Legendre's_conjecture comment "Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n2 and (n + 1)2 for every positive integer n. The conjecture is one of Landau's problems (1912) and remains unsolved.The prime number theorem suggests the actual number of primes between n2 and (n + 1)2 OEIS A014085 is about n/ln(n), i.e. about as many as the number of primes less than or equal to n.If Legendre's conjecture is true, the gap between any two successive primes would be .".
- Legendre's_conjecture label "Congettura di Legendre".
- Legendre's_conjecture label "Conjectura de Legendre".
- Legendre's_conjecture label "Conjecture de Legendre".
- Legendre's_conjecture label "Conjetura de Legendre".
- Legendre's_conjecture label "Legendre's conjecture".
- Legendre's_conjecture label "Legendresche Vermutung".
- Legendre's_conjecture label "Vermoeden van Legendre".
- Legendre's_conjecture label "勒讓德猜想".
- Legendre's_conjecture sameAs Legendresche_Vermutung.
- Legendre's_conjecture sameAs Conjetura_de_Legendre.
- Legendre's_conjecture sameAs Legendreren_aieru.
- Legendre's_conjecture sameAs Conjecture_de_Legendre.
- Legendre's_conjecture sameAs Congettura_di_Legendre.
- Legendre's_conjecture sameAs Vermoeden_van_Legendre.
- Legendre's_conjecture sameAs Conjectura_de_Legendre.
- Legendre's_conjecture sameAs m.07s8fl.
- Legendre's_conjecture sameAs Q1812503.
- Legendre's_conjecture sameAs Q1812503.
- Legendre's_conjecture sameAs Legendre's_conjecture.
- Legendre's_conjecture wasDerivedFrom Legendre's_conjecture?oldid=603264332.
- Legendre's_conjecture depiction Plot_of_number_of_primes_between_consecutive_squares.png.
- Legendre's_conjecture isPrimaryTopicOf Legendre's_conjecture.
