# Primality of number 1 [duplicate]

Is number 1 prime as per the definition of prime numbers? Because as per the definition for being prime it should be divided only by 1 and number itself.

• – lhf
Commented Dec 11, 2014 at 23:22

As in @Thekwasti's comment, the "question" of whether $1$ is a prime or not has no genuine mathematical content, but is only a question of convention. Indeed, for a long time it was considered a prime "by definition", and the assertions about unique factorization were all the more clumsy because of that. Thus, historically, there was a collective ... probably not overt... decision to declare $1$ not prime, to make the statements of the well-known theorems cleaner.

The decisive perspective on the issue is that our collective choice of terminology does not change the underlying reality.

No. For a number to be prime, it must have exactly two divisors. $1$ has only one divisor.

It's important for $1$ not to be a prime, because one of the most important theorems about prime numbers is this:

Every number can be written as the product of prime numbers in only one way (not counting different orders).

Examples: $15=3\times5$, and that is the only way to write $15$ as the product of primes.
$150=2\times3\times5\times5$, and that is the only way to write $150$ as the product of primes.

This wouldn't be true if $1$ were a prime. If it were, then $15$ could be written as $3\times5$, $1\times3\times5$, $1\times1\times3\times5$, etc.

This means that, if "prime" were defined in such a way to include $1$, then the primes would be a lot less useful. Therefore, we exclude $1$ from the primes.

The number 1 is not a prime because it is defined that way. It is really important because if 1 was a prime number, there would be no unicity of an integer decomposition into prime number factors.

I am so old I was taught that definition, that it's enough for a prime number to be divisible only by $1$ and itself. But my children and grandchildren were taught that a prime number must have exactly two distinct divisors among the positive integers. When I first heard about this redefinition, I thought it made sense.

Because, you see, there were some things that bothered me about $1$ being a prime number, some things that didn't quite make sense to me. For example, consider the sequence $p^n$ where $n$ runs over the positive integers. $p^n$ gets larger as $n$ gets larger. But $1^n = 1$ no matter what $n$ is. In this way, $1$ is different from the prime numbers. But it is also different from the composite numbers.

There is at least one property $1$ shares with some composite numbers: $\sqrt{1} \in \mathbb{Q}$. But $\sqrt[n]{1} = 1$ no matter what $n$ is. Compare a number like $4$, see that while $\sqrt{4} = 2$, $\sqrt[3]{4} \not\in \mathbb{Q}$.

As a schoolchild I wasn't able to express these misgivings I had about $1$ being a prime number. But it was clear to me that $1$ differs from the prime numbers in a much more fundamental way than $2$ differs from the odd primes.

• Thank you! No hand-wringing over the FTA, just facts about 1. Commented Dec 12, 2014 at 3:15
• Then again 1 satisfies several properties usually reserved for prime numbers, such as Wilson's theorem. Commented Dec 12, 2014 at 15:37
• It's not that those properties are somehow "reserved" for the prime numbers. A number either satisfies a property or it doesn't. You can find a lot of properties satisfied by 1 and the prime numbers (see Sloane's A008578 for some of them). But you can also find lots of properties that 2 and the odd primes satisfy but 1 doesn't. Commented Dec 14, 2014 at 0:40

By the definition of a prime, a prime is divisible by 2 numbers: 1 AND itself. Because 1 is only divisible by 1 number, 1, 1 is not a prime.

No, the number 1 is not a prime. Prime numbers are usually explicitly defined to be natural numbers larger than 1 so that the number is only divisible by 1 and the number it self.

• But why the definition says about being divisible from the number 1 and the number itself? Isn't this definition true for Number 1
– Sumit Jaiswal
Commented Dec 8, 2014 at 14:56
• My definition says "a positive integer $n$ is called prime if and only if it is divisible by exactly $2$ distinct positive integers". It is still a matter of definition, but for most (all that I'm aware of) applications, excluding $1$ makes sense. This also conforms nicely to generalizations of the concept of primality such as, for instance, in ring theory.
– user193271
Commented Dec 8, 2014 at 15:03
• It is true that the number $1$ is only divisible by $1$ and it self. However, primes are usually defined explicitly not to be $1$. This is just a convention adopted in math. Why this is the definition used I can not say, it may mainly be for convenience. For example, prime factorizations would not be unique if $1$ was also included in the primes.
– Guut Boy
Commented Dec 8, 2014 at 15:05
• In earlier times $1$ was considered a prime number. But like the example for unique decomposition of integers into prime numbers it turned out, that in almost every theorem considering primes $1$ has to be excluded (like "Let $p$ be a prime but not $1$. Then ...). I guess this is one of the reasons the definition has changed. Commented Dec 8, 2014 at 19:49

From the Wikipedia article on prime numbers:

Most early Greeks did not even consider $1$ to be a number, and so they did not consider it a prime. In the 19th century, however, many mathematicians did consider the number $1$ a prime. For example, Derrick Norman Lehmer's list of primes up to $10{,}006{,}721$, reprinted as late as 1956, started with $1$ as its first prime. Henri Lebesgue is said to be the last professional mathematician to call $1$ prime.

The point is, whether you consider $1$ to be a prime -- that is, whether you define "prime number" in such a way that $1$ meets the definition -- is merely a matter of convention. The modern convention excludes it. There seems to be general agreement (vide other answers here) that this is done to simplify various assertions involving primes, which would otherwise require the phrase "greater than $1$." (Note, we do consign ourselves to utilizing the phrase "greater than $2$" or the adjective "odd" in quite a few settings where primes appear.) The price we pay is putting that phrase in elementary statements such as "Every number greater than $1$ is either prime or composite."

Considering $$1$$ a prime muddies the notion of being coprime or relatively prime. Although $$a$$ and $$b$$ are coprime or relatively prime if $$\gcd (a,b)=1$$, this relationship is in fact derivative of the more basic notion (from which the words actually arise) that $$a$$ and $$b$$ are coprime or relatively prime when they have no common prime factors. If $$1$$ is a prime, that more basic notion of sharing no prime factors has to be forever after qualified with the explicit exception "other than $$1$$."