Bernoulli numbers
E141076
Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Bernoulli numbers canonical | 4 |
| Faulhaber's formula | 1 |
| Faulhaber’s formula for sums of powers of integers | 1 |
| OEIS:A164555 (Bernoulli numbers B_n) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1233824 — 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: Bernoulli numbers Context triple: [Jakob Bernoulli, knownFor, Bernoulli numbers]
-
A.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
B.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
C.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
D.
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.
-
E.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
- 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: Bernoulli numbers Target entity description: Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
-
A.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
B.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
C.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
D.
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.
-
E.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical constant sequence
ⓘ
sequence of rational numbers ⓘ |
| alsoKnownAs | B_n ⓘ |
| application |
analytic continuation of zeta-type functions
ⓘ
evaluation of sums of integer powers ⓘ |
| B_0 | 1 ⓘ |
| B_1 | -1/2 ⓘ |
| B_10 | 5/66 ⓘ |
| B_12 | -691/2730 ⓘ |
| B_2 | 1/6 ⓘ |
| B_3 | 0 ⓘ |
| B_4 | -1/30 ⓘ |
| B_5 | 0 ⓘ |
| B_6 | 1/42 ⓘ |
| B_7 | 0 ⓘ |
| B_8 | -1/30 ⓘ |
| B_9 | 0 ⓘ |
| connectedTo |
Kummer’s criterion for Fermat’s Last Theorem
ⓘ
linked to:
Kummer congruences
irregular primes ⓘ |
| definedAs | coefficients in the Taylor expansion of x/(e^x - 1) ⓘ |
| definitionFormula | x/(e^x - 1) = \sum_{n=0}^{\infty} B_n x^n / n! ⓘ |
| field |
mathematical analysis
ⓘ
number theory ⓘ |
| firstAppearance | Ars Conjectandi ⓘ |
| generalization |
Bernoulli polynomials
ⓘ
p-adic Bernoulli numbers ⓘ |
| generatingFunction | t/(e^t - 1) ⓘ |
| growthProperty | absolute values grow roughly like (2n)!/(2\pi)^{2n} ⓘ |
| indexDomain | nonnegative integers ⓘ |
| introducedBy |
Jacob Bernoulli
ⓘ
linked to:
Jakob Bernoulli
|
| notablePrime | 691 divides numerator of B_12 ⓘ |
| parityProperty | B_n = 0 for all odd n > 1 ⓘ |
| recurrenceRelation | \sum_{k=0}^{n} {n+1 \choose k} B_k = 0 for n \ge 1 ⓘ |
| relatedTo |
Bernoulli polynomials
ⓘ
Riemann zeta function ⓘ Stirling numbers ⓘ |
| sequenceID |
OEIS:A027641 (even-indexed Bernoulli numbers)
ⓘ
OEIS:A164555 (Bernoulli numbers B_n) ⓘ
linked to:
Bernoulli numbers
|
| signPattern | even-indexed Bernoulli numbers alternate in sign ⓘ |
| usedIn |
Euler–Maclaurin summation formula
ⓘ
Faulhaber’s formula for sums of powers of integers ⓘ
linked to:
Bernoulli numbers
asymptotic expansions of special functions ⓘ expansion of log(\Gamma(x)) ⓘ expansion of the Riemann zeta function at negative integers ⓘ series for the cotangent function ⓘ series for the tangent function ⓘ |
| valueType | rational numbers ⓘ |
| yearOfIntroduction | 1713 ⓘ |
| zetaRelation | \zeta(1-n) = -B_n/n for n \ge 1 ⓘ |
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: Bernoulli numbers Description of subject: Bernoulli numbers are a sequence of rational numbers that play a central role in number theory and analysis, especially in formulas for sums of powers of integers and in the study of special functions like the Riemann zeta function.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Bernoulli numbers
linked to: Bernoulli numbers
linked to: Bernoulli numbers