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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 < kn 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.


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

[math]\displaystyle{ 0 \lt a_1 \lt a_2 \cdots \lt a_{\phi(n)} \lt n , }[/math]

the mean square gap satisfies

[math]\displaystyle{ \sum_{i=1}^{\phi(n)-1} (a_{i+1}-a_i)^2 \lt C n^2 / \phi(n) }[/math]

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

See also


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

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