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.
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 ⋯ < a ϕ ( n ) < n , {\displaystyle 0<a_{1}<a_{2}\cdots <a_{\phi (n)}<n,}
the mean square gap satisfies
∑ i = 1 ϕ ( n ) − 1 ( a i + 1 − a i ) 2 < C n 2 / ϕ ( n ) {\displaystyle \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.
See also Reduced residue system
References
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.