Why is one not prime

It becomes clearer with examples that the dichotomy of prime and composite should in fact be a trichotomy of prime, composite and unit.

For any positive counting number, there is a unique monotonically non-decreasing list of primes whose product is that number; this is its prime factorisation; the multiplicity of each prime as a factor of the number is the number of times the prime shows up in this list. However, indeed, 0 does not have a prime factorisation, much less a unique one.

It has every prime as a factor, as many times as you like. Indeed, if you consider the least common multiple — normally at least as big as each factor — of any number and zero, the answer is zero; while the highest common factor — usually no greater than any of its multiples — of any number and zero is the given number. In this sense, zero is greater than all other whole numbers; for multiplicative purposes, zero is infinity, the product of arbitrarily many of each and every prime all at once.

It is a better game when 1 is not a prime, because the unique factorization theorem is so much fun. Skip to content. About Highlights Events Links. Since time immemorial, humans have asked: Why are we here? What is our purpose? Using this definition, 1 can be divided by 1 and the number itself, which is also 1, so 1 is a prime number. However, modern mathematicians define a number as prime if it is divided by exactly two numbers. It is important to remember that mathematical definitions develop and evolve.

Throughout history, many mathematicians considered 1 to be a prime number although that is not now a commonly held view. So when debating if 1 is a prime number, I'm prepared to call it a draw. If 2 is prime, -2 should be as well. I assiduously avoided defining prime in the previous paragraph because of an unfortunate fact about the definition of prime when it comes to these larger sets of numbers: it is wrong!

In the positive whole numbers, each prime number p has two properties:. The number p cannot be written as the product of two whole numbers, neither of which is a unit. The first of these properties is what we might think of as a way to characterize prime numbers, but unfortunately the term for that property is irreducible.

The second property is called prime. In the case of positive integers, of course, the same numbers satisfy both properties. So 2 is irreducible, but it is not prime. In this set of numbers, 6 can be factored into irreducible numbers in two different ways. But there are similar number sets that have an infinite number of units. As sets like this became objects of study, it makes sense that the definitions of unit, irreducible, and prime would need to be carefully delineated.

In particular, if there are number sets with an infinite number of units, it gets more difficult to figure out what we mean by unique factorization of numbers unless we clarify that units cannot be prime.

While I am not a math historian or a number theorist and would love to read more about exactly how this process took place before speculating further, I think this is one development Caldwell and Xiong allude to that motivated the exclusion of 1 from the primes.

As happens so often, my initial neat and tidy answer for why things are the way they are ended up being only part of the story. Thanks to my friend for asking the question and helping me learn more about the messy history of primality. The views expressed are those of the author s and are not necessarily those of Scientific American. Sometimes numbers with this property are called irreducible and then the name prime is reserved for those numbers which when they divide a product ab , must divide a or b these classes are the same for the ordinary integers--but not always in more general systems.

Nevertheless, the units are a necessary precursors to the primes, and one falls in the class of units, not primes. FAQ: Why is the number one not prime? Answer One: By definition of prime! The definition is as follows. An integer greater than one is called a prime number if its only positive divisors factors are one and itself.

Clearly one is left out, but this does not really address the question "why? Using the definition above he proved: The Fundamental Theorem of Arithmetic Every positive integer greater than one can be written uniquely as a product of primes, with the prime factors in the product written in order of nondecreasing size.

Answer Three: Because one is a unit.