See rank of apparition in All languages combined, or Wiktionary
Download JSON data for rank of apparition meaning in English (4.2kB)
{
"forms": [
{
"form": "ranks of apparition",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {
"1": "ranks of apparition"
},
"expansion": "rank of apparition (plural ranks of apparition)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Number theory",
"orig": "en:Number theory",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"_dis": "50 50",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w+disamb"
}
],
"examples": [
{
"ref": "1986, Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer, page 114",
"text": "Let (W#x5F;n) be an EDS associated to an elliptic curve E#x2F;K and a nonzero point P#x5C;inE(K) of finite order. Let r#x5C;ge 2 be the smallest index such that W#x5F;r#x3D;0. (The number r is called the rank of apparition of the sequence.)\n[…]\nSuppose that K is a finite field and that the rank of apparition r of (W#x5F;n) is at least 4.",
"type": "quotation"
},
{
"text": "1996, D. L. Wells, Residue Counts Modulo Three for the Fibonacci Triangle, G. E. Bergum, Andreas N. Philippou, Alwyn F. Horadam (editors), Applications of Fibonacci Numbers, Kluwer Academic, Softcover reprint, page 535,\nA similar identification between Pascal's Triangle modulo p and the Fibonacci Triangle modulo p can be made for primes p which have the length of the period equal to twice their rank of apparition in the Fibonacci Sequence."
},
{
"ref": "2013, E. L. Roettger, H. C. Williams, R. K. Guy, “Some Extensions of the Lucas Functions”, in Jonathan M. Borwein, Igor Shparlinski, Wadim Zudilin, editors, Number Theory and Related Fields, Springer, page 303",
"text": "Indeed, as shown in [18, Theorem 4.27], there exist sequences #x5C;#x7B;U#x5F;n#x5C;#x7D; and primes p for which p has three ranks of apparition. In the previous section, we showed that if p#x3D;2, then p has no more than two ranks of apparition in #x5C;#x7B;U#x5F;n#x5C;#x7D;.",
"type": "quotation"
}
],
"glosses": [
"Given a positive integer m and a divisibility sequence Sₖ, the smallest index k such that Sₖ is divisible by m;\nsuch an index for some generalisation of the concept (for example to allow multiple ranks of apparition for a given m).",
"Given a positive integer m and a divisibility sequence Sₖ, the smallest index k such that Sₖ is divisible by m;"
],
"id": "en-rank_of_apparition-en-noun-ZiyG3lFr",
"links": [
[
"number theory",
"number theory"
],
[
"positive",
"positive"
],
[
"integer",
"integer"
],
[
"divisibility sequence",
"divisibility sequence"
]
],
"raw_glosses": [
"(number theory) Given a positive integer m and a divisibility sequence Sₖ, the smallest index k such that Sₖ is divisible by m;\n"
],
"synonyms": [
{
"english": "for the [[Fibonacci sequence]]",
"word": "Fibonacci entry point"
}
],
"topics": [
"mathematics",
"number-theory",
"sciences"
]
},
{
"categories": [
{
"kind": "topical",
"langcode": "en",
"name": "Number theory",
"orig": "en:Number theory",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"_dis": "50 50",
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w+disamb"
}
],
"glosses": [
"Given a positive integer m and a divisibility sequence Sₖ, the smallest index k such that Sₖ is divisible by m;\nsuch an index for some generalisation of the concept (for example to allow multiple ranks of apparition for a given m).",
"such an index for some generalisation of the concept (for example to allow multiple ranks of apparition for a given m)."
],
"id": "en-rank_of_apparition-en-noun-bamKCdUQ",
"links": [
[
"number theory",
"number theory"
],
[
"positive",
"positive"
],
[
"integer",
"integer"
],
[
"divisibility sequence",
"divisibility sequence"
]
],
"raw_glosses": [
"(number theory) Given a positive integer m and a divisibility sequence Sₖ, the smallest index k such that Sₖ is divisible by m;\n"
],
"synonyms": [
{
"english": "for the [[Fibonacci sequence]]",
"word": "Fibonacci entry point"
}
],
"topics": [
"mathematics",
"number-theory",
"sciences"
]
}
],
"word": "rank of apparition"
}
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English lemmas",
"English multiword terms",
"English nouns"
],
"forms": [
{
"form": "ranks of apparition",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {
"1": "ranks of apparition"
},
"expansion": "rank of apparition (plural ranks of apparition)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
"English terms with quotations",
"en:Number theory"
],
"examples": [
{
"ref": "1986, Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer, page 114",
"text": "Let (W#x5F;n) be an EDS associated to an elliptic curve E#x2F;K and a nonzero point P#x5C;inE(K) of finite order. Let r#x5C;ge 2 be the smallest index such that W#x5F;r#x3D;0. (The number r is called the rank of apparition of the sequence.)\n[…]\nSuppose that K is a finite field and that the rank of apparition r of (W#x5F;n) is at least 4.",
"type": "quotation"
},
{
"text": "1996, D. L. Wells, Residue Counts Modulo Three for the Fibonacci Triangle, G. E. Bergum, Andreas N. Philippou, Alwyn F. Horadam (editors), Applications of Fibonacci Numbers, Kluwer Academic, Softcover reprint, page 535,\nA similar identification between Pascal's Triangle modulo p and the Fibonacci Triangle modulo p can be made for primes p which have the length of the period equal to twice their rank of apparition in the Fibonacci Sequence."
},
{
"ref": "2013, E. L. Roettger, H. C. Williams, R. K. Guy, “Some Extensions of the Lucas Functions”, in Jonathan M. Borwein, Igor Shparlinski, Wadim Zudilin, editors, Number Theory and Related Fields, Springer, page 303",
"text": "Indeed, as shown in [18, Theorem 4.27], there exist sequences #x5C;#x7B;U#x5F;n#x5C;#x7D; and primes p for which p has three ranks of apparition. In the previous section, we showed that if p#x3D;2, then p has no more than two ranks of apparition in #x5C;#x7B;U#x5F;n#x5C;#x7D;.",
"type": "quotation"
}
],
"glosses": [
"Given a positive integer m and a divisibility sequence Sₖ, the smallest index k such that Sₖ is divisible by m;\nsuch an index for some generalisation of the concept (for example to allow multiple ranks of apparition for a given m).",
"Given a positive integer m and a divisibility sequence Sₖ, the smallest index k such that Sₖ is divisible by m;"
],
"links": [
[
"number theory",
"number theory"
],
[
"positive",
"positive"
],
[
"integer",
"integer"
],
[
"divisibility sequence",
"divisibility sequence"
]
],
"raw_glosses": [
"(number theory) Given a positive integer m and a divisibility sequence Sₖ, the smallest index k such that Sₖ is divisible by m;\n"
],
"synonyms": [
{
"english": "for the [[Fibonacci sequence]]",
"word": "Fibonacci entry point"
}
],
"topics": [
"mathematics",
"number-theory",
"sciences"
]
},
{
"categories": [
"English terms with quotations",
"en:Number theory"
],
"glosses": [
"Given a positive integer m and a divisibility sequence Sₖ, the smallest index k such that Sₖ is divisible by m;\nsuch an index for some generalisation of the concept (for example to allow multiple ranks of apparition for a given m).",
"such an index for some generalisation of the concept (for example to allow multiple ranks of apparition for a given m)."
],
"links": [
[
"number theory",
"number theory"
],
[
"positive",
"positive"
],
[
"integer",
"integer"
],
[
"divisibility sequence",
"divisibility sequence"
]
],
"raw_glosses": [
"(number theory) Given a positive integer m and a divisibility sequence Sₖ, the smallest index k such that Sₖ is divisible by m;\n"
],
"synonyms": [
{
"english": "for the [[Fibonacci sequence]]",
"word": "Fibonacci entry point"
}
],
"topics": [
"mathematics",
"number-theory",
"sciences"
]
}
],
"word": "rank of apparition"
}
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-06 from the enwiktionary dump dated 2024-05-02 using wiktextract (f4fd8c9 and c9440ce). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.