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.

This entry was posted in -- By the Mathematician, Math, Number Theory. Bookmark the permalink.

32 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. 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.

5. 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.

6. 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.)

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

8. 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?

9. 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.

10. sushma says:

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

11. bbqiy says:

this is a very complicated question to think of, but the answer is very simple.

For example , all prime numbers should have one factors, but 1 only has 1, although some people will think or 1 as 1×1.If 1 is a prime number, then people wounld be able to do this:

the factor of 6 is 3x2x1x1x1x1x1x1x1x1x1x1x1

in conclusion 1 cannot be a prime number.End of case

12. Aaron says:

I agree with Jeremy. It appears that the exclusion of the number 1 is relatively arbitrary. There are many formulae that rely on the multiplication of ordered primes which are then diffed with primes not included in the multiplication. These formulae work nicely using 1 or any other primes as the differencing factors. I would have to say that I believe ignoring the power of a +/- P (including 1 in the set of P) would be premature and somewhat random.

13. Yugolkishor says:

1 is not a prime number because the product of any amount of 1 will be 1.
For example : 1×1=1
1x1x1=1
1x1x1x1=1
1x1x1x1x1=1
and so on .

14. farah says:

Well, I think, if it includes 1 as the prime number, then, every number could actually divide by itself or to the others. Which, I mean, every number have a factor of 1. How can we differentiate a number with a prime numbers, if every numbers are actually have a factor of 1.

15. omaish says:

1- by defination prime nos are greater than 1
2- all prime numbers are not perfect squares while 1 is a perfect square
3- all prime numbers have two divisors while 1 has only 1 divisor

16. franc says:

great mathematicians you better think of having 1 as a prime number. i had a class in one of the universities in kenya. i taught that….then the question is why is one not a prime number.

17. Desirae says:

I still dont understand why #1 is not a prime number?

18. Sanni Ibraheem Alabi says:

Hi, my undergraduate research work was on primality(i.e determining whether a given number is prime or not), so let me say this: ” The #1 was taken to be prime for a long time but was later removed explicitly. Reason being that many useful theorems involving primes were ‘collapsing’ if 1 were prime. E.g Fermat’s Little Theorem, Wilson Theorem etc. Hence, if u take a cursory look at d definition of prime number, u may be forced to include 1(A prime number is an integer p such that p is divisible exactly by +p, -p, +1 or -1). But mathematicians intentionally excluded 1 from primes in modern day advanced number theory, for convenience & for continuity. Another justification is that 1 is undoubtedly a perfect square, whereas a number cannot be a perfect square and be a prime at d same time. In conclusion, out of every general rule, there are exceptions. 1 is an exception in prime definition. Thanks.

19. julia says:

simply the answer is because one has no factors so it cannot be put to any category

20. John Matthews says:

So poor one is like Pluto. It’s still the same but left out because someoe changed the definition

21. Because 1 is divisible by 1 which IS itself so its not divisible by one and itself it’s divisible by one which is itself… So overlooked XD

22. Tanu says:

Because 1 is divisible by 1 which IS itself so its not divisible by one and itself it’s divisible by one which is itself… So overlooked XD

23. Tanu says:

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 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

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.

24. ramya says:

1 is not a prime number because 1divides all number
but 2 is prime number which is divides 1 and itself
3 is prime number which is divides 1 and itself
5 is prime number which is divides 1 and itself
7 is prime number which divides 1 and itself
and so on

25. chethan says:

Prime number is a number with exactly two divisors. In this way 1 is not a prime number.

26. David causey says:

Dear Respondents, is it not so that one is no longer considered to be prime simply because we have decided that it is not? The definition has changed over the years. Nobody is asking whether it is. They want to know why it is not: The answer is that it is more convenient to say it is not.
The classic definition was a prime was divisible only by one and itself. One clearly satisfies that definition

The recent definition of primes having exactly two divisors came about to make computerized problem solving less time consuming. Strictly a useful tool, modified to suit the times.

27. Jozef S. says:

So having read the above, I conclude that the exclusion of 1 is rather arbitrary. Simply, for convenience sake. However, philosophically, 1 has also prime number properties, it is divisible by 1 and itself. We want the definition to facilitate a larger spectrum of prime numbers theory rather than to attempt addressing “prime number properties of 1” which in consequence would hinder a number of useful theowrong.Please Correct me if I’m wrong.

28. marie says:

hi how u doin?

29. marie says:

haha!

30. The Physicist says:

@marie
Real well, how you doin’ marie?

31. Bobby says:

I disagree with the idea that this is all about “definitions”. I think 1 was never prime, and the earlier inclusion of it was a mistake.

The purest method of finding primes, indeed the way they were initially defined, is via a sieve technique. Ignoring one for a moment, write down the numbers 2 through 100. Circle the lowest number (2), this is the first prime. Cross of all multiples of 2. Then circle the lowest number not yet circled or crossed off (3). Cross off all multiples of 3 not already crossed off. Etc. At the end your circled numbers are the primes below 100.

Now try that again, but include 1. You circle that, then cross off every multiple of 1. Epic fail at this point, you just crossed off the rest of the list.

1 is not prime.