Q: Why is the number 1 not considered a prime number?

Mathematician: Note that when we say that a number is “prime”, all that we are doing is applying a definition that was devised by mathematicians. A prime number is generally defined to be any positive number that has exactly two distinct positive integer divisors (the divisors being 1 and the number itself). So 13 is prime, because it is divisible only by 1 and 13, whereas 14 is not prime because it is divisible by 1, 2, 7 and 14. Note that this excludes the number 1 from being prime. The biggest reason this definition of primality is used, as opposed to a slightly different one, is merely a matter of convenience. Mathematicians like to choose definitions in such a way that important theorems are simple and easy to state. Probably the most important theorem involving prime numbers is the Fundamental Theorem of Arithmetic, which says that all integers greater than 1 can be expressed as a unique product of prime numbers up to reordering of the factors. So, for example, 54 can be written as  54 = 3*3*3*2 which is a unique factorization assuming that we list the factors in decreasing order. Now, notice that if we counted 1 as a prime number, then this theorem would no longer hold as stated, since we would then be able to write

54 = 3*3*3*2*1 = 3*3*3*2*1*1 = 3*3*3*2*1*1*1

so there would not be a single, unique representation for 54 as the theorem requires. Hence, if we count 1 as a prime number, then the Fundamental Theorem ofArithmetic would have to be restated as something like, “all integers greater than 1 can be expressed as a unique product of prime numbers (not including 1) up to reordering of the factors.” This is a tiny bit more cumbersome, but not horrible. If you have to work with prime numbers day in and day out though, simplifying theorems just a little bit (by choosing your definitions carefully) may well be worth it. Nonetheless, if mathematicians chose a slightly different definition for primality that included the number one, while they would then be forced to modify many of their theorems involving primes, the world wouldn’t come crashing down on its head.

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11 Responses to Q: Why is the number 1 not considered a prime number?

  1. Andy says:

    1 x 1 = 1 thus 1 is the identity number for primes, multiplication, division etc…..simples!
    keep up the good work peeps…..

  2. Nitika says:

    Because when we consider 1 it would not satisfy the fundamental theorem of airthematics

  3. Pingback: 1 is not a prime number « lchemm2

  4. Pingback: Why 1 is not a prime number « Math 300 – Writing for Mathematics. Fall 2012

  5. ewrrwer says:

    Stuff have definitions.
    An example is even. The definition of even number is: “An even number is an integer that is “evenly divisible” by 2″

    1 is not prime because of the definition of prime.
    Prime is “An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself”
    So to a number be prime it need to be greater than one.
    The question is why someone would think 1 is prime, if the definition of prime say that’s impossible? maybe those guys forget the part “greater than one” at prime definition.

  6. starscape says:

    If one is a prime number than so would the other sqaured numbers but that can’t be right because the other squared numbers ex: 4,9,16,25, ect …. is divisible by other numbers that’s not one. So 1 is a square root and a squared number but not a prime number. Let’s pretend that one was a prime number because 1 is a factor of 1: 1×1=1 and if 1 is a squared number then why can’t 4 be a prime number? The answer is simple, because 4 has the factor of 1,2, and 4 so for the sake of the other squared number 1 simply cannot be a prime number.

  7. Jeremy says:

    ewrrwer, I think don’t think people who ask this question are ignorant about the definition of prime numbers. They know that the definition specifically excludes 1 and wonder whywhy only consider numbers greater than 1?

    Most people think of a prime number as being divisible only by 1 and itself. Well, 1 is divisible only by 1 and itself. It does seem rather arbitrary to add an exception just to get rid of 1.

    In fact, the exclusion of 1 from the prime numbers is relatively new. For centuries, 1 was indeed considered alongside 2, 3, 5, etc. As the article says, it just becomes more convenient in advanced number theory if 1 is excluded. That’s how “greater than 1″ entered the definition.

    (Alternatively, the primes can be defined, like in the article, as numbers with exactly two factors. To me, this alternative definition makes it a little less tempting to include 1.)

  8. Oladele says:

    So simple,base on the definition,and i dont even think there should be futher expalanation.

  9. Mikkel says:

    the whole “all integers greater than 1 can be expressed as a unique product of prime numbers up to reordering of the factors” doesn’t apply to prime numbers does it?

  10. Why 1 is not considered a prime number ?
    If a number X have only two positive divisors, 1 and X where X is not equal to 1.In case of 1 there are only one divisor i.e. 1, so by definition 1 is not considered a prime number.

  11. sushma says:

    Prime number should have two factors, but 1 have only one factor

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