Overview: What Are Prime Numbers?
The definition of a prime number is any number greater than one that can only be divided evenly by 1 and itself.. Therefore, 2 is a prime number because its only factors are 1 and 2, and 3 is a prime number because its only factors are 1 and 3. The number 4 is not because it has another factor than 1 and 4, the number 2. It is a composite number.
Why Are Prime Numbers Important?
Prime numbers are the building blocks of the natural (counting) numbers, and they are considered an important part of number theory. One mathematician proposed a theorem that every even number greater than 2 is the sum of two primes, which has neither been proven or disproven. In fact, they are considered so essential that mathematicians have sent signals into space that are sequences of prime numbers, in the hopes that other intelligent civilizations would respond to the signal. (So far, there have been no responses.) They have applications in computer science, and are the basis for high-level codes.
Why Isn’t One a Prime Number?
The number 1 is neither a prime number nor a composite number because it doesn’t have all the properties of either primes or composites. In fact, the ancient Greeks, who discovered prime numbers, didn’t consider the number 1 to be a number, so it wasn’t an element of their definitions. One of the important theorems in arithmetic is that every number can be expressed as a unique product of primes, and there are many other arguments that follow, only if the number one is not part of the definition.
Finding Prime Numbers
The ancient Greeks, such as Euclid, used prime numbers in the development of number theory. They theorized that the set of prime numbers was infinite, since the set of numbers is infinite, and there can always be one number greater. Smaller numbers are found by methods of trial division, although that method has too many variables for very large numbers. Computers use a different algorithm for large numbers that involve exponents and another theorem.
Finding Prime Factors
If every number (over 1) can be expressed as an unique product of prime numbers, what does that mean? For one thing, it means that one can factor numbers completely to find prime factors. Looking at the number 24, it can be factored as 3 X 8. Three is a prime number, but 8 is not. Eight is the product of 2 X 2 X 2. The prime factorization of 24 can be read as 24 = 3 X 2 X 2 X 2. Prime factors can be found by repeated division, and students can factor into prime numbers using a factor tree. (For fun, since the number 24 is an even number greater than 2, the 2 prime numbers added together that equal 24 are 5 and 19.)
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