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<div>In [[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]] to&nbsp;{{mvar|n}}. [[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|multiplicative group of integers modulo ''n'']].<br />
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The distribution of totatives has been a subject of study. [[Paul Erdős]] conjectured that, writing the totatives of ''n'' as<br />
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:<math> 0 < a_1 < a_2 \cdots < a_{\phi(n)} < n ,</math><br />
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the mean square gap satisfies<br />
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:<math> \sum_{i=1}^{\phi(n)-1} (a_{i+1}-a_i)^2 < C n^2 / \phi(n) </math><br />
<br />
for some constant ''C'' and this was proved by [[Bob Vaughan]] and [[Hugh Montgomery (mathematician)|Hugh Montgomery]].<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. (2) | volume=123 | pages=311–333 | year=1986 }}</ref><br />
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==See also==<br />
*[[Reduced residue system]]<br />
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==References==<br />
{{reflist}}<br />
* {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 | at=B40 }}<br />
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==Further reading==<br />
*{{Citation | last=Sándor | first=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 }}<br />
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==External links==<br />
*{{MathWorld |title=Totative |id=Totative}}<br />
*{{PlanetMath |urlname=Totative |title=totative}}<br />
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[[Category:Modular arithmetic]]<br />
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