# I Real reason why 1 isn't prime?

1. Dec 27, 2016

### FallenApple

It's very interesting that primes are considered the atoms of numbers, but yet, 1 isn't prime.

It makes sense because one isn't an "atom" for numbers in the multiplication sense. That is,
6=2*3
=1*2*3
=1*1*2*3
=1*.....
So clearly 1 cannot be prime.

But in the addition sense, all numbers are made of ones. So in the addition sense, one is an "atom" for numbers.

But multiplication is just a special case of addition.

Is there something deeper going on here?

I know that by definition, 1 isn't prime. But what is the philosophical justification?

Could it be that the concept of a one as a basic unit has already been encapsulated by primes? That is, a prime is already made of differing amounts of ones(by repeated addition). All composite numbers are made of primes. All primes are made of just ones.

So primes are kinda like the atoms of numbers while 1 is just the quark that makes up the primes. Could this be a valid viewpoint?

2. Dec 27, 2016

### BvU

So is 0 a prime number according to your definition ?
No

3. Dec 27, 2016

### dkotschessaa

1 not being prime is, I think, a sort of convention we use because considering it as such would not lead to anything interesting, since everything has 1 as a divisor. So we include in the requirement for primes that they be greater than one. In other words, having 1 as a prime doesn't gain one anything, and would probably make other proofs and definitions unnecessarily complicated. Possibly this isn't a very satisfying justification, but some definitions are like this.

I think it is valid to wonder (well, at least I have wondered) why we "favor" multiplication when we talk about primes being the "building blocks of numbers." Certainly they only "build" numbers in the multiplicative sense. What about addition?

For some reason prime numbers get all the glory in this sense, but number theorists and combinatorialists do talk about ways to build numbers additively by considering integer partitions. i.e. (example from URL)

4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1

Are all partitions of 4. It gets exceedingly difficult to find all the partitions when $n$ is large.

There is validity to your statement:

This is actually recognized in the idea of a generator. For the additive group of positive integers, 1 is a generator.

-Dave K

4. Dec 27, 2016

### Staff: Mentor

In general it's: A unit isn't prime. And the definition of a prime number is $p\,\vert \,a\cdot b \Rightarrow p\,\vert \,a \vee p\,\vert \,b$. This is trivially fulfilled for units so it makes no sense to require it for them. If we'd allow units to be prime, then thousands of statements had to explicitly rule them out, which is unnecessarily inconvenient. I think the basis of this concept has been the fundamental theorem of arithmetic in which an arbitrary number of $\pm 1$ is simply ugly and a ballast. The absence of units allow many other statements to be written in a nice way.

5. Dec 27, 2016

### Staff: Mentor

We could include 1 in the prime numbers. It is purely a convention that we did not.

Currently many proofs have statements like "for every prime, ...", with a changed definition most of them would have to be "for every prime apart from 1, ...". That is just unnecessary. "For every prime and for 1, ...", where the redefinition would make it easier, rarely occurs.

6. Dec 27, 2016

### Staff: Mentor

One holds a special place as the multiplicative identity and so by definition is a factor of all numbers.

7. Dec 27, 2016

### FactChecker

If we called 1 a prime, many of the statements we make about primes would have to include a special exception for 1.

8. Dec 30, 2016

As per my knowledge, 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).

If one is a prime number than so it would the other squared 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.

9. Dec 30, 2016

### jbriggs444

For the ring of integers, that definition is as good as any.

But, by the above definition it is not. You should not assume that it is. Anything follows from a contradiction. Including arbitrary facts about squareness.

Last edited: Dec 30, 2016
10. Dec 30, 2016

Staff Emeritus
First, the thread title seems to suggest that there is a real reason that students of mathematics are not being told. This is not only nonsense, it's a conspiracy theory.

Second, "kinda" is not a word. It's not even slang. It's baby talk. If you want people to take you seriously, you should avoid it.

Third, there are four kinds of numbers: composites, primes, units and zeros. This is a definition, and thus a convention, but it is a useful convention. We could use other conventions, but this would force us into awkward statements like "for all primes not one" or "for all composites with more than two factors". This is no different than what is done in other fields: we could call plants and animals by the same name, but then we would be stuck with expressions like "those sessile animals with leaves."

11. Dec 30, 2016

### Dr. Courtney

Good thinking always separates DEFINITIONS and PRESUPPOSITIONS from results following from those based on the logic of that thinking.

Whether or not 1 is a prime is a DEFINITION.

It is not a consequence of logic.

Considering primes as the "atoms of numbers" is simply a descriptive analogy. It may be useful or not, but it is not prescriptive for whether a number belongs to the set of primes.

ZERO and ONE and i are special. They often need their own categories. Like Einstein and Newton.

12. Dec 30, 2016

### lavinia

For commutative rings primes are defined differently. A number is prime if whenever it divides the product of two numbers it divides one of them.

In an integral domain a number is called irreducible if whenever it is the product of two numbers then one of those numbers is a unit.

For the ring of integers prime and irreducible are the same but for other integral domains they can be different. A prime is always irreducible but an irreducible may not be prime.

One might say that a unit in a ring is a prime but it is a trivial case since units divide every number.

13. Jan 2, 2017

May be you are right at your own way, but as per my knowledge the logic is right. It proves 1 is not prime. Still you want any related proofs you can visit this. http://alienryderflex.com/even_prime.shtml
After reading this and other books, I have find this logic. If you have other solution for this you can share with me. It would be grateful.

14. Jan 2, 2017

### symbolipoint

No claim of being an expert on this, but if you factorize this composite, just an example,

28=4*7
4*7=2*2*1*7*1
4*7=1*2*2*1*7*1

4*7=1*1*2*2*1*7
4*7=1*1*1*2*2*7

4*7=1*1*2*2*7
4*7=1*2*2*7
4*7=2*2*7

The repeated or very few, or lack of showing the "1" factor does not change the meaning of the product. The occurances of the "2" or the "7" is important, and fewer or more of either of them WILL change the product.

15. Jan 3, 2017

### Peter Martin

I think the definition of "prime number" includes the requirement that it be divisible only by itself and 1. In the case of 1, these two numbers are the single number, namely 1. The word "and" implies more than one thing, so that eliminates 1 as a prime number.

16. Jan 3, 2017

### symbolipoint

What I am trying to point is that you can include more than multiplicity 1 for the factor of 1 and this will not change the product. Maybe the very clear and too easily missed logic from Peter Martin is the real reason that 1 is not prime.

17. Jan 4, 2017

### Warp

Whether 1 is considered a prime or not is a matter of agreement by the mathematical community. It was agreed at some point that it's significantly more convenient, for quite many reasons, to exclude it from the list of primes. It significantly minimizes the amount of exceptions (of the form "all primes except 1") having to be made in all sorts of theorems and proofs, and makes many such things much leaner, concise and logical. (The Fundamental Theorem of Arithmetic would be your quintessential example.)

So 1 is not a prime, nor is it a composite. It has its own category.

18. Jan 5, 2017

### Stephen Tashi

Yes, that's one answer to "why". The problem with this sort of thread is that the question "Why ...?." is ambiguous.

On the one hand we have "Why...?" in the sense of a request for an logical or axiomatic justification. In that case it must be understood that definitions are arbitrary stipulations.

On the other hand, we have "Why...?" in the sense of "Why did Molly steal the candy?". To say "Molly sole the candy because her actions satisfied the definition of 'stealing candy' probably isn't the kind of answer a questioner has in mind. Inquiring about "Why isn't 1 prime?" can be a question about human motivation and behavior. We can consider the sociological or psychological question of "Why did mathematicians choose to write the definition of prime so that 1 was excluded?".

It helps if people who answer of this type of question made it clear which sense of "why" they are answering. Answering a question like "Why isn't 1 prime?" or "Why can't we divide by zero?" etc. by trying to give a mathematical "proof" amounts to trying to prove a definition, which encourages a gross misunderstanding of how mathematics is organized.

An argument that 1 is not prime should not be presented as a mathematical proof. It should be presented as motivation for certain human behavior.

It's tempting to use a proof-by-contradiction approach to justify human behavior. We are tempted to say "Suppose we define 1 to be prime , then...." or "Suppose we could divide by zero. Then let 1/0 = k for some number k. ..." In an intuitive sense, we can certainly think along those lines. However, it would be extremely difficult to formulate this type of reasoning as legitimate mathematics. It involves a scenario where we need one sort of axiomatic system (e.g one where "assume we define 1 to be a prime" is a defined statement) to write proofs about a different axiomatic system (e.g. standard number theory where we do not define 1 to be a prime). Sophisticated logicians might be able to formulate definitions and axioms for a system of mathematics that proves theorems about other systems of mathematics. However, when we fail to that, we should make it clear that such arguments are not formal proofs.

Last edited: Jan 5, 2017
19. Jan 5, 2017

### FactChecker

The important point. If you want unique factorization into primes, you don't want to call 1 a prime.

20. Jan 5, 2017

### symbolipoint

I believe that is what I am trying to say in giving my example.