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) |

(2) |

(3) | |||

(4) |

Solving for gives

(5) |

(6) |

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

(7) |

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).

**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.

© 1996-9

1999-05-26