Triple

T4552364
Position Surface form Disambiguated ID Type / Status
Subject Hardy–Littlewood circle method E120394 entity
Predicate appliedTo P1129 FINISHED
Object Waring's problem
Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
E451523 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Waring's problem | Statement: [Hardy–Littlewood circle method, appliedTo, Waring's problem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Waring's problem
Context triple: [Hardy–Littlewood circle method, appliedTo, Waring's problem]
  • A. Lagrange's four-square theorem
    Lagrange's four-square theorem is a fundamental result in number theory stating that every natural number can be expressed as the sum of four integer squares.
  • B. Fermat polygonal number theorem
    The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
  • C. 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.
  • D. Hardy–Littlewood circle method
    The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
  • E. Hardy–Littlewood conjectures
    The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Waring's problem
Triple: [Hardy–Littlewood circle method, appliedTo, Waring's problem]
Generated description
Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Waring's problem
Target entity description: Waring's problem is a famous conjecture in number theory that concerns representing natural numbers as sums of fixed powers of integers and determining how many such powers are needed.
  • A. Lagrange's four-square theorem
    Lagrange's four-square theorem is a fundamental result in number theory stating that every natural number can be expressed as the sum of four integer squares.
  • B. Fermat polygonal number theorem
    The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
  • C. 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.
  • D. Hardy–Littlewood circle method
    The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
  • E. Hardy–Littlewood conjectures
    The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69bd4636f1648190a701445c2fcd9c17 completed March 20, 2026, 1:05 p.m.
NER Named-entity recognition batch_69bd57f7b9748190af29d02fc77b02e0 completed March 20, 2026, 2:21 p.m.
NED1 Entity disambiguation (via context triple) batch_69bdb95b01b0819094a600752e41aa09 completed March 20, 2026, 9:17 p.m.
NEDg Description generation batch_69bdbdbf73508190b64a78ff9274ee6d completed March 20, 2026, 9:35 p.m.
NED2 Entity disambiguation (via description) batch_69bdbe1bcd8c819094adea59c91c6f5b completed March 20, 2026, 9:37 p.m.
Created at: March 20, 2026, 1:09 p.m.