“Composite Number”的概念、定义、习题解题方法及翻译-数学辞典

单词

Composite Number

释义

Composite Number

A Positive Integer which is not Prime (i.e., which has Factors other than 1 and itself).

A composite number can always be written as a Product in at least two ways (since is always possible). Callthese two products

(1)

then it is obviously the case that

(

divides

). Set

(2)

where

is the part of

which divides

, and

the part of

which divides

. Then there are

and

such that

			(3)
			(4)

Solving

for

gives

(5)

It then follows that



			(6)

It therefore follows that

is never Prime! In fact, the more general result that

(7)

is never Prime for

an Integer

also holds (Honsberger 1991).

There are infinitely many integers of the form and which are composite, where is the Floor Function (Forman and Shapiro, 1967; Guy 1994, p. 220). The first few composite occurfor , 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, ... (Sloane's A046037), and the few composite occur for , 8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, ... (Sloane's A046038).

See also Amenable Number, Grimm's Conjecture, Highly Composite Number, Prime FactorizationPrime Gaps, Prime Number

References

Forman, W. and Shapiro, H. N. ``An Arithmetic Property of Certain Rational Powers.'' Comm. Pure Appl. Math. 20, 561-573, 1967.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.

Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 19-20, 1991.

Sloane, N. J. A. Sequences A002808/M3272, A046037, and A046038 in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.

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