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单词 EulerPhiFunction
释义

Euler phi function


For any positive integer n, φ(n) is the number of positive integers less than or equal to n which are coprimeMathworldPlanetmath to n. The function φ is known as the φ function. This function may also be denoted by ϕ.

Among the most useful of φ are the facts that it is multiplicative (meaning if gcd(a,b)=1, then φ(ab)=φ(a)φ(b)) and that φ(pk)=pk-1(p-1) for any prime p and any positive integer k. These two facts combined give a numeric computation of φ for all positive integers:

φ(n)=np|n(1-1p).(1)

For example,

φ(2000)=2000p|2000(1-1p)
=2000(1-12)(1-15)
=2000(12)(45)
=800010
=800.

From equation (1), it is clear that φ(n)n for any positive integer n with equality holding exactly when n=1. This is because

p|n(1-1p)1,

with equality holding exactly when n=1.

Another important fact about the φ function is that

d|nφ(d)=n,

where the sum extends over all positive divisorsMathworldPlanetmathPlanetmathPlanetmath of n. Also, by definition, φ(n) is the number of units in the ring /n of integers modulo n.

The sequence {φ(n)} appears in the OEIS as sequence http://www.research.att.com/ njas/sequences/A000010A000010.

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更新时间:2025/5/3 20:26:36