## 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). Call these 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 occur for , 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 Factorization Prime 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.