Composite Number
A composite number is defined as a positive integer greater than 1 that is not prime, meaning it has factors other than 1 and itself. For example, the integer 6 is composite because it can be factored as 1 × 6 and also as 2 × 3. In contrast, prime numbers are those that can only be expressed as a product of 1 and themselves. Numbers like 4 and 9 are also composite, being factored as 2 × 2 and 3 × 3, respectively. Composite numbers can be categorized into subtypes, including semiprimes, which have exactly two prime factors; sphenic numbers, which are composed of three distinct prime factors; and powerful numbers, which can be factored into squares or cubes. The understanding of composite numbers is essential in number theory and has practical applications in fields such as cryptography. Overall, recognizing composite numbers aids in comprehending the structure of integers and their relationships.
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Composite Number
Positive integers are commonly referred to as the counting, natural, or whole numbers. This set of integers comprises only the positive whole numbers and is denoted as Z+. By considering the fundamental theorem of arithmetic, these numbers, when greater than 1, can be classified as either prime or a combination of multiplying prime numbers. Simply put, prime numbers are the product between only two factors. For instance, the positive integer 2 is a prime number, because it is the product of 1 × 2. The number 1 is an exception and is neither a prime nor a composite number.
The composite numbers refer to positive integers that can be factored as any combination of other integers besides 1 and themselves. For instance, 9 is a composite number as it can be the result of the product 3 × 3. Composite numbers are part of the number theory and along with prime numbers are widely employed in modern cryptography.
Overview
A composite number is a positive integer
, greater than 1 and which is not a prime (that is, has factors other than 1 and itself). For instance, the positive integer 6 is a composite number since it can be factored as 1 × 6 and also as 2 × 3. Thus, for positive integers
if there are other positive integers, where
, such that
, then
is a composite number.
Instead of considering them as a product of other positive integers, composite numbers can be observed from a different perspective by considering the fact that it has a prime divisor. For instance, the positive integer 4 can be divided by the prime number 2.
Mathematically speaking, a composite number
can be expressed as a product in at least two ways:
Considering that
divides
, and
, where
and
are the parts of
that divide
and
, respectively, we have
and
such that
and
.
By solving
for
, we have:
It then follows that a composite number is given by
This proof has been observed that a positive integer is never a prime for all powers
, expressed by
Composite numbers can be further categorized by the number of prime factors comprised as semiprime, sphenic, and powerful numbers. A semiprime number has only two factors besides 1 and itself, and one of these factors can be squared, for example, 6 (2 × 3) and 12 (22 × 3 ). A sphenic, or square-free number, is a composite number that has three distinct prime factors, for example, 30 = 2 × 3 ×5, and not divisible by a perfect square. Finally, the powerful numbers are composite numbers that can be factored as squares and/or cubes, for example, 108 = 22 × 33.
Bibliography
Caldwell, C., et al. "The history of the Primality of One—A Selection of Sources." Journal of Integer Sequences 15.2 (2012): 3.
Caldwell, Chris K., and Yeng Xiong. "What Is the Smallest Prime?" Journal of Integer Sequences 15.2 (2012): 3.
Ferguson, Niels, Bruce Schneier, and Tadayoshi Kohno. Cryptography Engineering: Design Principles and Practical Applications. Wiley, 2012.
Rosen, Kenneth. Elementary Number Theory and Its Applications. 6th ed. Pearson, 2010.
Wells, David. Prime Numbers: The Most Mysterious Figures in Math. Wiley, 2011.
Wu, Hung-Hsi, and Hongxi Wu. Understanding Numbers in Elementary School Mathematics. American Mathematical Soc., 2011.