Triple
T10991355
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | prime number theorem |
E259759
|
entity |
| Predicate | predecessorResult |
P96494
|
FINISHED |
| Object |
Chebyshev’s estimates for π(x)
Chebyshev’s estimates for π(x) are 19th-century bounds on the prime-counting function that showed it grows on the order of x/log x and provided a crucial precursor to the prime number theorem.
|
E898473
|
NE FINISHED |
How this triple was built (5 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: Chebyshev’s estimates for π(x) | Statement: [prime number theorem, predecessorResult, Chebyshev’s estimates for π(x)]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Chebyshev’s estimates for π(x) Context triple: [prime number theorem, predecessorResult, Chebyshev’s estimates for π(x)]
-
A.
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.
-
B.
Linnik’s theorem on the least prime in an arithmetic progression
Linnik’s theorem on the least prime in an arithmetic progression is a result in analytic number theory that gives an explicit upper bound, depending only on the modulus, for the size of the smallest prime in any given coprime residue class.
-
C.
Legendre’s conjecture on primes between consecutive squares
Legendre’s conjecture on primes between consecutive squares is an unproven statement in number theory asserting that there is always at least one prime number between any two consecutive perfect squares.
-
D.
Siegel–Walfisz theorem
The Siegel–Walfisz theorem is a result in analytic number theory that gives strong uniform estimates for the distribution of prime numbers in arithmetic progressions with relatively small moduli.
-
E.
Turán–Kubilius inequality
The Turán–Kubilius inequality is a fundamental result in probabilistic number theory that provides bounds on the distribution of additive arithmetic functions.
- 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: Chebyshev’s estimates for π(x) Triple: [prime number theorem, predecessorResult, Chebyshev’s estimates for π(x)]
Generated description
Chebyshev’s estimates for π(x) are 19th-century bounds on the prime-counting function that showed it grows on the order of x/log x and provided a crucial precursor to the prime number theorem.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Chebyshev’s estimates for π(x) Target entity description: Chebyshev’s estimates for π(x) are 19th-century bounds on the prime-counting function that showed it grows on the order of x/log x and provided a crucial precursor to the prime number theorem.
-
A.
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.
-
B.
Linnik’s theorem on the least prime in an arithmetic progression
Linnik’s theorem on the least prime in an arithmetic progression is a result in analytic number theory that gives an explicit upper bound, depending only on the modulus, for the size of the smallest prime in any given coprime residue class.
-
C.
Legendre’s conjecture on primes between consecutive squares
Legendre’s conjecture on primes between consecutive squares is an unproven statement in number theory asserting that there is always at least one prime number between any two consecutive perfect squares.
-
D.
Siegel–Walfisz theorem
The Siegel–Walfisz theorem is a result in analytic number theory that gives strong uniform estimates for the distribution of prime numbers in arithmetic progressions with relatively small moduli.
-
E.
Turán–Kubilius inequality
The Turán–Kubilius inequality is a fundamental result in probabilistic number theory that provides bounds on the distribution of additive arithmetic functions.
- F. None of above. chosen
PD
Predicate disambiguation
gpt-5-mini-2025-08-07
Target predicate: predecessorResult Context triple: [prime number theorem, predecessorResult, Chebyshev’s estimates for π(x)]
-
A.
predecessorFunction
Indicates a function that maps each element to its immediate predecessor in an ordered sequence or structure.
-
B.
predecessorStatus
Indicates that one entity’s status is determined by, or directly follows from, the status of a preceding entity in a sequence or process.
-
C.
predecessorOperator
Indicates that one operator precedes another in an ordered sequence or process.
-
D.
predecessorOperation
Indicates that one operation occurs or must be completed before another operation in a defined sequence.
-
E.
predecessor
Indicates that one entity comes before another in an ordered sequence or succession.
- F. None of above. chosen
Provenance (7 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_69d6aa8a6a548190a750f944ccdc8064 |
completed | April 8, 2026, 7:20 p.m. |
| NER | Named-entity recognition | batch_69d795d1e918819090c71f5a077fa15a |
completed | April 9, 2026, 12:04 p.m. |
| NED1 | Entity disambiguation (via context triple) | batch_69e34504ebec8190a78e4795765b0c24 |
completed | April 18, 2026, 8:47 a.m. |
| NEDg | Description generation | batch_69e3556fd3548190a33f04604be947cf |
completed | April 18, 2026, 9:57 a.m. |
| NED2 | Entity disambiguation (via description) | batch_69e3593b0f8481909ed7a90f8bb9839d |
completed | April 18, 2026, 10:13 a.m. |
| PD | Predicate disambiguation | batch_69d72e93ac648190b46c5d12bf3eb1e9 |
completed | April 9, 2026, 4:44 a.m. |
| PDg | Predicate description generation | batch_69d732242fdc8190be77d1f730a42935 |
completed | April 9, 2026, 4:59 a.m. |
Created at: April 8, 2026, 9:24 p.m.