Ramanujan prime
E355434
A Ramanujan prime is a type of prime number that provides a bound guaranteeing the existence of a certain number of primes in intervals of the form (x/2, x], named after the mathematician Srinivasa Ramanujan.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Ramanujan prime canonical | 1 |
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
type of prime number ⓘ |
| approximation | R_n ~ p_{2n} ⓘ |
| category |
eponymous mathematical objects
ⓘ
prime numbers ⓘ |
| classification | special prime ⓘ |
| condition | for all real x ≥ R_n ⓘ |
| countingIndex | n ⓘ |
| definition | the nth Ramanujan prime R_n is the least integer such that for all x ≥ R_n, π(x) − π(x/2) ≥ n ⓘ |
| field | number theory ⓘ |
| fifthTerm | 41 ⓘ |
| firstTerm | 2 ⓘ |
| fourthTerm | 29 ⓘ |
| generalizes | Bertrand's postulate to intervals containing at least n primes ⓘ |
| growthRate | R_n is asymptotic to the 2n-th prime ⓘ |
| guarantees | at least n primes in (x/2, x] ⓘ |
| hasIndexing | n = 1, 2, 3, ... ⓘ |
| inequality |
R_n < p_{3n}
ⓘ
R_n > p_{2n} ⓘ |
| infinitelyMany | true ⓘ |
| intervalForm | (x/2, x] ⓘ |
| introducedBy | Srinivasa Ramanujan ⓘ |
| isA | infinite sequence of prime numbers ⓘ |
| namedAfter | Srinivasa Ramanujan ⓘ |
| OEISSequence | A104272 ⓘ |
| property |
R_n is strictly increasing with n
ⓘ
R_n ≥ 2n for all n ≥ 1 ⓘ each Ramanujan prime is itself a prime number ⓘ guarantees at least n primes in the interval (x/2, x] for all x ≥ R_n ⓘ |
| publishedIn | Proceedings of the London Mathematical Society ⓘ |
| relatedConcept |
Chebyshev functions
ⓘ
surface form:
Chebyshev function
prime distribution in short intervals ⓘ prime gap ⓘ |
| relatedFunction | prime-counting function π(x) ⓘ |
| relatedTo | Bertrand's postulate ⓘ |
| researchArea | distribution of primes in dyadic intervals ⓘ |
| secondTerm | 11 ⓘ |
| sequenceBegins | 2, 11, 17, 29, 41, 47, 59, 67, 71, 97 ⓘ |
| subfield | analytic number theory ⓘ |
| symbol | R_n ⓘ |
| thirdTerm | 17 ⓘ |
| usedIn |
bounds on primes in short intervals
ⓘ
refinements of Bertrand-type results ⓘ |
| yearIntroduced | 1919 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.