# Totative

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{{Short description|Coprime number less than a given integer}}
In [number theory](/source/number_theory), a '''totative''' of a given positive integer {{mvar|n}} is an integer {{mvar|k}} such that {{math|0 < ''k'' ≤ ''n''}} and {{mvar|k}} is [coprime](/source/coprime) to&nbsp;{{mvar|n}}.  [Euler's totient function](/source/Euler's_totient_function) φ(''n'') counts the number of totatives of ''n''.  The totatives under multiplication modulo ''n'' form the [multiplicative group of integers modulo ''n''](/source/Multiplicative_group_of_integers_modulo_n).

==Distribution==
The distribution of totatives has been a subject of study.  [Paul Erdős](/source/Paul_Erd%C5%91s) conjectured that, writing the totatives of ''n'' as

:<math> 0 < a_1 < a_2 \cdots < a_{\phi(n)} < n ,</math>

the mean square gap satisfies

:<math> \sum_{i=1}^{\phi(n)-1} (a_{i+1}-a_i)^2 < C n^2 / \phi(n) </math>

for some constant ''C'', and this was proven by [Bob Vaughan](/source/Bob_Vaughan) and [Hugh Montgomery](/source/Hugh_Montgomery_(mathematician)).<ref>{{cite journal | doi=10.2307/1971274 | zbl=0591.10042 | last1=Montgomery | first1=H.L. | author1-link=Hugh Montgomery (mathematician) | last2=Vaughan | first2=R.C. | author2-link=Bob Vaughan | title=On the distribution of reduced residues | journal=Ann. Math. |series=2 | volume=123 | pages=311–333 | year=1986 | issue=2 | jstor=1971274 }}</ref>

==See also==
*[Reduced residue system](/source/Reduced_residue_system)

==References==
{{reflist}}
* {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[Springer-Verlag](/source/Springer-Verlag) |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 | at=B40 }}

==Further reading==
*{{Citation | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | zbl=1079.11001 | pages=242–250 }}

==External links==
*{{MathWorld |title=Totative |id=Totative}}
*{{PlanetMath |urlname=Totative |title=totative}}

Category:Modular arithmetic

{{Numtheory-stub}}

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