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 |
How this entity was disambiguated
This entity first appeared as the object of triple T3410517 — 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: Ramanujan prime Context triple: [Srinivasa Ramanujan, notableWork, Ramanujan prime]
-
A.
prime number theorem
The prime number theorem is a fundamental result in number theory that describes how prime numbers become less frequent and provides an approximate formula for the number of primes less than a given large number.
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B.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
-
C.
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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D.
Fermat number
A Fermat number is a special type of integer of the form \(F_n = 2^{2^n} + 1\), studied in number theory for its intriguing properties related to primality and constructible polygons.
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E.
Prime
Prime is one of the traditional canonical hours in the Christian liturgy of the hours, historically recited in the early morning.
- 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: Ramanujan prime Target entity description: 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.
-
A.
prime number theorem
The prime number theorem is a fundamental result in number theory that describes how prime numbers become less frequent and provides an approximate formula for the number of primes less than a given large number.
-
B.
Hardy–Ramanujan asymptotic formula
The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
-
C.
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.
-
D.
Fermat number
A Fermat number is a special type of integer of the form \(F_n = 2^{2^n} + 1\), studied in number theory for its intriguing properties related to primality and constructible polygons.
-
E.
Prime
Prime is one of the traditional canonical hours in the Christian liturgy of the hours, historically recited in the early morning.
- F. None of above. chosen
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 ⓘ |
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: Ramanujan prime Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.