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totative

{{Short description|A coprime number less than a given integer}}

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 {{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.

Distribution

The distribution of totatives has been a subject of study. Paul Erdős conjectured that, writing the totatives of n as

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

the mean square gap satisfies

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

for some constant C, and this was proven by Bob Vaughan and Hugh Montgomery.{{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 }}

See also

References

{{reflist}}

  • {{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 }}

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

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