Totative

In number theory, a totative of a given positive integer $n$ is an integer $k$ such that $0 < k \leq n$ 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.

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}<Cn^{2}/\phi (n)

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

See also

Reduced residue system

References

↑ Montgomery, H.L.; Vaughan, R.C. (1986). "On the distribution of reduced residues". Ann. Math. 2. 123: 311–333. doi:10.2307/1971274. Zbl 0591.10042.

Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. B40. ISBN 978-0-387-20860-2. Zbl 1058.11001.

Further reading

Sándor, Jozsef; Crstici, Borislav (2004), Handbook of number theory II, Dordrecht: Kluwer Academic, pp. 242–250, ISBN 1-4020-2546-7, Zbl 1079.11001

External links

Weisstein, Eric W. "Totative". MathWorld.
totative at PlanetMath.org.

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