“Divisor”的概念、定义、习题解题方法及翻译-数学辞典

单词

Divisor

释义

Divisor

A divisor of a number is a number which Divides , also called a Factor. The total number ofdivisors for a given number can be found as follows. Write a number in terms of its Prime Factorization

(1)

For any divisor

where

(2)

(3)

Now,

, so there are

possible values. Similarly, for

, there are

possible values, so the total number of divisors

is given by

(4)

The function

is also sometimes denoted

. The product of divisors can be found by writingthe number

in terms of all possible products

(5)


			(6)

and

(7)

The Geometric Mean of divisors is

(8)

The sum of the divisors can be found as follows. Let

with

and

. For any divisor

, where

is a divisor of

and

is a divisor of

. The divisors of

are 1,

,..., and

. The divisors of

are 1,

, ...,

. The sums of the divisors are then

(9)

(10)

For a given

(11)

Summing over all

(12)

. Splitting

and

into prime factors,

(13)

For a prime Power

, the divisors are 1,

, ...,

, so

(14)

For

, therefore,

(15)

For the special case of

a Prime, (15) simplifies to

(16)

For

a Power of two, (15) simplifies to

(17)

The Arithmetic Mean is

(18)

The Harmonic Mean is

(19)

But

, so

and

(20)

and we have

(21)

(22)

Given three Integers chosen at random, the probability that no common factor will divide them all is

(23)

where

is Apéry's Constant.

Let be the number of elements in the greatest subset of such that none of its elements are divisible by two others. For sufficiently large,

(24)

(Le Lionnais 1983, Lebensold 1976/1977).

See also Aliquant Divisor, Aliquot Divisor, Aliquot Sequence, Dirichlet Divisor Problem,Divisor Function, e-Divisor, Exponential Divisor, Greatest Common Divisor, InfinaryDivisor, k-ary Divisor, Perfect Number, Proper Divisor, Unitary Divisor

References

Guy, R. K. ``Solutions of .'' §B18 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 73-75, 1994.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 43, 1983.

Lebensold, K. ``A Divisibility Problem.'' Studies Appl. Math. 56, 291-294, 1976/1977.

随便看

数学辞典收录了8975条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。