Totative

Short description: A coprime number less than a given integer

In number theory, a totative of a given positive integer n is an integer k such that 0 < k ≤ n and k is coprime to 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.

## Contents

* 1 Distribution
* 2 See also
* 3 References
* 4 Further reading
* 5 External links

## Distribution

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

0<a1<a2⋯<aϕ(n)<n,

the mean square gap satisfies

∑i=1ϕ(n)−1(ai+1−ai)2<Cn2/ϕ(n)

for some constant C, and this was proven by Bob Vaughan and Hugh Montgomery.[1]

## See also

* Reduced residue system

## References

1. ↑ Montgomery, H.L.; Vaughan, R.C. (1986). "On the distribution of reduced residues". Ann. Math.. 2 123: 311–333. doi:10.2307/1971274.

* Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. B40. ISBN 978-0-387-20860-2.

## Further reading

* Sándor, Jozsef; Crstici, Borislav (2004), Handbook of number theory II, Dordrecht: Kluwer Academic, pp. 242–250, ISBN 1-4020-2546-7

## External links

* Weisstein, Eric W.. "Totative". http://mathworld.wolfram.com/Totative.html.
* totative at PlanetMath.org.

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