Composite Number

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

A composite number $C$ can always be written as a Product in at least two ways (since $1\cdot C$ is always possible). Call these two products

$$C=ab=cd,$$ (1)

then it is obviously the case that $C\vert ab$ ($C$ divides $ab$). Set

$$c=mn,$$ (2)

where $m$ is the part of $C$ which divides $a$, and $n$ the part of $C$ which divides $n$. Then there are $p$ and $q$ such that

$$a = mp$$ (3)

$$b = nq.$$ (4)

Solving $ab=cd$ for $d$ gives

$$d={ab\over c}={(mp)(nq)\over mn}=pq.$$ (5)

It then follows that

$$S \equiv a^2+b^2+c^2+d^2$$

$$= m^2p^2+n^2q^2+m^2n^2+p^2q^2$$

$$= (m^2+q^2)(n^2+p^2).$$ (6)

It therefore follows that $a^2+b^2+c^2+d^2$ is never Prime! In fact, the more general result that

$$S\equiv a^k+b^k+c^k+d^k$$ (7)

is never Prime for $k$ an Integer $\geq 0$ also holds (Honsberger 1991).

There are infinitely many integers of the form $\left\lfloor{(3/2)^n}\right\rfloor$ and $\left\lfloor{(4/3)^n}\right\rfloor$ which are composite, where $\left\lfloor{x}\right\rfloor$ is the Floor Function (Forman and Shapiro, 1967; Guy 1994, p. 220). The first few composite $\left\lfloor{(3/2)^n}\right\rfloor$ occur for $n=8$, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, ... (Sloane's A046037), and the few composite $\left\lfloor{(4/3)^n}\right\rfloor$ occur for $n=5$, 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.