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In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
, a totative of a given positive integer is an integer such that and is
coprime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
to .
Euler's totient function In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and numbe ...
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 Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a renowned Hungarian mathematician In this page we keep the names in Hungarian order (family name first). {{compact ToC , short1, side=yes A * Alexits György (1899–1 ...

Paul Erdős
conjectured that, writing the totatives of ''n'' as : 0 < a_1 < a_2 \cdots < a_ < n , the mean square gap satisfies : \sum_^ (a_-a_i)^2 < C n^2 / \phi(n) for some constant ''C'', and this was proven by
Bob Vaughan Robert Charles "Bob" Vaughan FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Fa ...
and Hugh Montgomery.


See also

*
Reduced residue system In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...


References

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Further reading

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External links

* * Modular arithmetic {{Numtheory-stub