See primitive root in All languages combined, or Wiktionary
Download JSON data for primitive root meaning in English (4.7kB)
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"form": "primitive roots",
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{
"ref": "1941, Derrick Henry Lehmer, Guide to Tables in the Theory of Numbers, National Research Council, page 13",
"text": "There are #x5C;phi(p-1) incongruent primitive roots of p. The fact that there are so many primitive roots causes no difficulty in the theory of the binomial congruence but has caused considerable confusion in the tabulation of primitive roots.",
"type": "quotation"
},
{
"ref": "1992, Joe Roberts, Lure of the Integers, Mathematical Association of America, page 55",
"text": "The integers 2, 3, 4, and 6 each have exactly one primitive root and therefore, by default, each has a set of primitive roots consisting of \"consecutive\" integers.\nThe integer 5, with primitive roots of 2 and 3 is the only positive integer having at least two primitive roots for which the entire set of primitive roots are consecutive integers.",
"type": "quotation"
},
{
"ref": "2006, Neville Robbins, Beginning Number Theory, Jones & Bartlett Learning, page 159",
"text": "For example, the prime 7 has #x5C;phi(6)#x3D;2 primitive roots, namely, 3 and 5. Also, the prime 11 has #x5C;phi(10)#x3D;4 primitive roots, namely, 2, 6, 7, 8.\nRecall from Theorem 6.7 that if m has primitive roots, and if g is one primitive root (#x5C;operatorname#x7B;mod#x7D;m), then we can obtain all primitive roots (#x5C;operatorname#x7B;mod#x7D;m) by raising g to appropriate exponents.",
"type": "quotation"
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"glosses": [
"For a given modulus n, a number g such that for every a coprime to n there exists an integer k such that gᵏ ≡ a (mod n); a generator (or primitive element) of the multiplicative group, modulo n, of integers relatively prime to n."
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"raw_glosses": [
"(mathematics, number theory) For a given modulus n, a number g such that for every a coprime to n there exists an integer k such that gᵏ ≡ a (mod n); a generator (or primitive element) of the multiplicative group, modulo n, of integers relatively prime to n."
],
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{
"word": "multiplicative order"
}
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{
"sense": "number that generates other numbers modulo n",
"word": "generator"
},
{
"sense": "number that generates other numbers modulo n",
"word": "primitive element"
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{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "yuángēn",
"sense": "number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element",
"word": "原根"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element",
"word": "primitiivinen juuri"
},
{
"code": "fr",
"lang": "French",
"sense": "number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element",
"tags": [
"feminine"
],
"word": "racine primitive"
},
{
"code": "de",
"lang": "German",
"sense": "number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element",
"tags": [
"feminine"
],
"word": "Primitivwurzel"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element",
"tags": [
"feminine"
],
"word": "frumstæð rót"
},
{
"code": "it",
"lang": "Italian",
"sense": "number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element",
"tags": [
"feminine"
],
"word": "radice primitiva"
}
],
"wikipedia": [
"primitive root"
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"word": "primitive root"
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{
"ref": "1941, Derrick Henry Lehmer, Guide to Tables in the Theory of Numbers, National Research Council, page 13",
"text": "There are #x5C;phi(p-1) incongruent primitive roots of p. The fact that there are so many primitive roots causes no difficulty in the theory of the binomial congruence but has caused considerable confusion in the tabulation of primitive roots.",
"type": "quotation"
},
{
"ref": "1992, Joe Roberts, Lure of the Integers, Mathematical Association of America, page 55",
"text": "The integers 2, 3, 4, and 6 each have exactly one primitive root and therefore, by default, each has a set of primitive roots consisting of \"consecutive\" integers.\nThe integer 5, with primitive roots of 2 and 3 is the only positive integer having at least two primitive roots for which the entire set of primitive roots are consecutive integers.",
"type": "quotation"
},
{
"ref": "2006, Neville Robbins, Beginning Number Theory, Jones & Bartlett Learning, page 159",
"text": "For example, the prime 7 has #x5C;phi(6)#x3D;2 primitive roots, namely, 3 and 5. Also, the prime 11 has #x5C;phi(10)#x3D;4 primitive roots, namely, 2, 6, 7, 8.\nRecall from Theorem 6.7 that if m has primitive roots, and if g is one primitive root (#x5C;operatorname#x7B;mod#x7D;m), then we can obtain all primitive roots (#x5C;operatorname#x7B;mod#x7D;m) by raising g to appropriate exponents.",
"type": "quotation"
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],
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"For a given modulus n, a number g such that for every a coprime to n there exists an integer k such that gᵏ ≡ a (mod n); a generator (or primitive element) of the multiplicative group, modulo n, of integers relatively prime to n."
],
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"(mathematics, number theory) For a given modulus n, a number g such that for every a coprime to n there exists an integer k such that gᵏ ≡ a (mod n); a generator (or primitive element) of the multiplicative group, modulo n, of integers relatively prime to n."
],
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"synonyms": [
{
"sense": "number that generates other numbers modulo n",
"word": "generator"
},
{
"sense": "number that generates other numbers modulo n",
"word": "primitive element"
}
],
"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "yuángēn",
"sense": "number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element",
"word": "原根"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element",
"word": "primitiivinen juuri"
},
{
"code": "fr",
"lang": "French",
"sense": "number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element",
"tags": [
"feminine"
],
"word": "racine primitive"
},
{
"code": "de",
"lang": "German",
"sense": "number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element",
"tags": [
"feminine"
],
"word": "Primitivwurzel"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element",
"tags": [
"feminine"
],
"word": "frumstæð rót"
},
{
"code": "it",
"lang": "Italian",
"sense": "number such that gk ≡ a (mod n) exists for every a coprime to n — see also generator, primitive element",
"tags": [
"feminine"
],
"word": "radice primitiva"
}
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"word": "primitive root"
}
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-01 from the enwiktionary dump dated 2024-04-21 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.
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