Prime Numbers
Prime numbers are natural numbers greater than 1 that have exactly two distinct positive divisors: 1 and the number itself. This unique property distinguishes them from composite numbers, which have multiple factors. For instance, while the number 12 has six factors (1, 2, 3, 4, 6, and 12) and is therefore composite, the number 13 can only be divided into factors of 1 and 13, making it a prime number. The process of breaking down numbers into their prime factors is known as prime factorization, which is fundamental in mathematics, as every integer greater than 1 can be uniquely expressed as a product of prime numbers.
Additionally, there are special classes of prime numbers, such as Mersenne primes, which are defined as numbers that are one less than a power of two. Mersenne primes have notable connections to perfect numbers, which are numbers equal to the sum of their proper divisors. The largest known prime number, discovered in December 2017, is a Mersenne prime, showcasing the ongoing exploration and significance of prime numbers in mathematical research. Understanding prime numbers is essential for various fields, including cryptography and number theory, highlighting their importance beyond basic mathematics.
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Prime Numbers
A prime number is any natural number, or non-negative integer, that is greater than 1 and has only two positive divisors, or factors: itself and 1. A factor or divisor is an integer that can be multiplied by another integer to produce a product. The only way to calculate a prime number p as the product of two integers is to multiply p itself by the number 1, as in the equation p × 1 = p (or 1 × p = p).
Prime numbers are related to factorization, which is the breaking down of an integer (or other mathematical object) into its various factors—that is, into smaller integers that, when multiplied together, produce the original integer. One way to understand factorization is to think of it as dividing one group of objects into multiple smaller groups of equal size. For example, a group of twelve cookies can be broken up into twelve groups of one cookie each (factor pair 12 × 1), six groups of two cookies each (6 × 2), four groups of three cookies each (4 × 3), three groups of four cookies each (3 × 4), two groups of six cookies each (2 × 6), or one group of twelve cookies (1 × 12). These are the only quantities in which all twelve cookies will be evenly distributed among all groups. Thus, the number 12 has six factors—1, 2, 3, 4, 6, and 12—and is not prime, but rather a composite number. On the other hand, a group of thirteen cookies can only be broken up into either thirteen groups of one cookie each (13 × 1) or one group containing all thirteen cookies (1 × 13); dividing the cookies into any other number of groups would require the groups to either be unequal or include fractions of a whole cookie. This means that 13 has only two factors, 1 and 13, and is therefore prime.
The goal of factorization is often to break down the original number into its smallest possible factors. In the case of integers, the smallest possible factors would be prime numbers (and 1), because these numbers cannot be factorized any further. This process is called prime factorization. Of the six factors of the number 12, only two of them—2 and 3—are prime. The factors that correspond to these prime factors, 6 and 4, are composite numbers and can be further factorized as 2 × 3 and 2 × 2, respectively. Note that breaking down the factor pairs 2 × 6 and 3 × 4 to prime factors produces identical factors (2, 2, and 3), simply in a different order. Thus, the prime factorization of 12 can be expressed as 12 = 2 × 2 × 3, or 12 = 22 × 3 for short.
The number 1 is disregarded in prime factorization because it is the empty product, meaning that multiplying it by another number has no effect on that number. It used to be considered a prime number, but this was changed because including 1 as a prime number would require exceptions to be made to many of the rules that apply to other prime numbers. In fact, considering 1 to be a prime number even violates the principle of unique factorization that is the basis of the law of quadratic reciprocity, which has also been called the fundamental theorem of arithmetic.
Simply put, the fundamental theorem of arithmetic states that every integer greater than 1 can be represented as the product of a unique number of occurrences of prime factors. In other words, 2 × 2 × 3 will always equal 12; the order of the two 2s and one 3 may change, but they cannot produce any other number, and no other prime numbers or differing quantities of 2s and 3s can ever be multiplied together to produce 12. The problem with considering 1 to be a prime number is that any whole number can be multiplied or divided by 1 an infinite number of times without changing, and therefore 1 could occur as a factor an infinite number of times. After all, if 2 × 2 × 3 produces 12, so does 2 × 2 × 3 × 1, and so does 2 × 2 × 3 × 1 × 1. The same will hold true if 1 occurs once, twice, twenty, or two hundred times. In this respect, the number 1 is fundamentally different from all other numbers.
There are certain classes, or subsets, of prime numbers that are grouped together because they share a certain property or are all generated by a certain formula. One such class is Mersenne primes, named for seventeenth-century French polymath Marin Mersenne. A Mersenne number is any number whose value is one less than a power of two, as represented by the formula Mn = 2n – 1; Mersenne primes are simply Mersenne numbers that are prime. If the n in the previous equation is a composite number, then the resulting Mersenne number will also be composite (i.e., not prime), but the reverse does not hold true; n may be a prime number, such as 11, and still produce a composite number (M11 = 2047, which has the factor pair 23 × 89). Mersenne primes are notable for the association with perfect numbers, or positive integers that are equal to the sum of their positive proper divisors (i.e., all divisors other than themselves): according to a proof by the ancient Greek mathematician Euclid, for every Mersenne prime 2n – 1, there is a perfect number of the formula 2n – 1(2n – 1). The largest prime number ever found, discovered in December 2017 by the internet-based Great Internet Mersenne Prime Search (GIMPS), is a Mersenne prime in which n equals 77,232,917.
Bibliography
Angel, Allen R., and Dennis C. Runde. Elementary Algebra for College Students. 9th ed., Pearson, 2015.
Katz, Victor J. A History of Mathematics: An Introduction. 3rd ed., Pearson, 2018.
McKellar, Danica. Kiss My Math: Showing Pre-Algebra Who’s Boss. Hudson Street Press, 2008.
Miller, Julie. College Algebra Essentials. McGraw-Hill, 2014.
Revell, Timothy. "Largest Prime Number Ever Found Has Over 23 Million Digits." New Scientist, 4 Jan. 2018, www.newscientist.com/article/2157773-largest-prime-number-ever-found-has-over-23-million-digits/. Accessed 12 Feb. 2018.
Strogatz, Steven. The Joy of x: A Guided Tour of Math, from One to Infinity. Houghton Mifflin Harcourt, 2012.
Tent, Margaret W. "Understanding the Properties of Arithmetic: A Prerequisite for Success in Algebra." Mathematics Teaching in the Middle School, vol. 12, no. 1, 2006, pp. 22–25.
Weisstein, Eric W. "Mersenne Prime." Wolfram MathWorld, mathworld.wolfram.com/MersennePrime.html. Accessed 12 Feb. 2018.