TITLE: Should we or should we not take $1$ as a prime number?
QUESTION [7 upvotes]: I think I know that there were times in the past when it was convenient to look at a number $1$ as a prime number, and, as far as I can remember, even then it was dependent on who we ask is it prime or not?
This question is primarily here because in some of the questions I posted in the last few days I wanted that we for the purpose of the question consider $1$ as prime, or because of some similar reason.
So, let us see why, in my opinion, we should take $1$ as prime, and why we should not.
First, the definition of prime numbers as seen on Wikipedia:

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

If we interpret here this greater as $n \in \mathbb N \setminus \{1\}$, in other words, as strictly greater, then clearly $1$ cannot be prime because it is, by definition, not taken into consideration.
If we change the definition and remove part of the sentence, the part greater than 1, then we have this definition:

A prime number (or a prime) is a natural number that has no positive divisors other than 1 and itself.

Then we could say that $1$ is prime, because he has no other positive divisors other than 1 and itself, and he has two divisors which coincide and are both equal to $1$ (1 and itself).
But do we stumble upon some problem if we declare $1$ as prime?
Yes we do, but all these problems (that I know of) are not so serious.
First, we should rephrase fundamental theorem of arithmetic which says that:

Every positive integer $n>1$ can be represented in exactly one way as a product of prime powers.

If we would take $1$ as prime then the fundamental theorem of arithmetic could be stated as:

Every positive integer $n>1$ can be represented in exactly one way as a product of prime powers in such a way that in the factorization of $n$ powers of number $1$ are absent.

This is because we have $1^a=1^b$ for $a \neq b$ so if we want unique factorization then powers of $1$ must be absent from the factorization.
But do we gain something positive if we declare $1$ as prime number?
I think we do.
For example, the Goldbach conjecture states:

Every even integer greater than 2 can be expressed as the sum of two primes.

If we would declare $1$ as prime we could shorten the Goldbach conjecture:

Every even natural number can be expressed as the sum of two primes.

Of course, this is because $1+1=2$ so now even two has a representation as a sum of two primes.
But in this case we would gain even more. Now, when $1$ is taken as a prime number, then we have slightly greater chance that every even natural number can be expressed as a sum of two primes because now we can consider sums of the form $2n=(2n-1)+1$ and we cannot consider them when we do not take $1$ as a prime.
It seems (but I could be wrong), that it is the matter of agreement between mathematicians should we or should we not take $1$ as a prime number.
So, I have a few questions:

1)  Is there any serious problem or inconsistency inside of mathematics if we take $1$ as a prime number (other than just re-writing the theorems/lemmas/conjectures/definitions in a different way)?
2)  Should we or should we not take $1$ as a prime number?
3)  Can you give some other examples of conjectures for which it would be better to take $1$ as a prime?
4)  What are your thoughts on this?

REPLY [3 votes]: 1) Is there any serious problem or inconsistency inside of mathematics if we take 1 as a prime number (other than just re-writing the theorems/lemmas/conjectures/definitions in a different way)?

No, no serious problem, only a bunch of minor but unnecessary inconveniences.

2) Should we or should we not take 1 as a prime number?

We should not. Unless we like making things more difficult than they need to be.

3) Can you give some other examples of conjectures for which it would be better to take 1 as a prime?

Well, no, but for constructing magic prime squares it's occasionally useful to accept 1 as a prime.
4) What are your thoughts on this?
The prime numbers come first and our theorems about them second. It's like in chemistry: a metal is a metal regardless of whether or not we recognize it as such. And so in mathematics, 1 and $-1$ have always been units, long before we recognized them as such.
We're in agreement that 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... are all primes, right? Just checking. What characteristics do they have in common? Does 1 share all those characteristics? No, it does not. In fact, there are also many characteristics 1 does not share with the composite numbers either. Therefore 1 is neither prime nor composite, but a unit.
That we recognize this does not change the nature of the number. 1 has always been a unit. It's quite a different question from whether Y is a vowel or a consonant. All of us English speakers could get together and decide it one way or the other. But if there is a civilization in some distant world, they may too at first consider 1 to be a prime number but gradually come to realize that it is not. I venture to guess the Vulcans realized this a lot sooner than the Klingons, wink, wink.