Why is 1 not classified as being a prime number ?
Why is 1 not classified as being a prime number ?
Hi!
I'd like to know why 1 is not classified as being a prime number, and how to really turn this fact down.
I already have an esquisse in large part (although brief in text) of my reasoning, but would like to keep an open question as the title, in a first time, before exposing my concordant arguments, to avoid the phenomenon [I only agree/I only disagree].
Thank to those who will expose their angle of view.
For consistency and coherence purpose, let's first give the definition of a prime number (I think to have it, but want be guaranteed that we have the same).
Thanks!
Write you later.
I'd like to know why 1 is not classified as being a prime number, and how to really turn this fact down.
I already have an esquisse in large part (although brief in text) of my reasoning, but would like to keep an open question as the title, in a first time, before exposing my concordant arguments, to avoid the phenomenon [I only agree/I only disagree].
Thank to those who will expose their angle of view.
For consistency and coherence purpose, let's first give the definition of a prime number (I think to have it, but want be guaranteed that we have the same).
Thanks!
Write you later.

 Posts: 3797
 Joined: Fri Oct 25, 2013 6:09 am
Re: Why is 1 not classified as being a prime number ?
A prime number has to be divisible by I and itself not just itself which is the case for I
But I have also heard that it is simply a convention of mathematics that I is not prime
But I have also heard that it is simply a convention of mathematics that I is not prime
Re: Why is 1 not classified as being a prime number ?
I thank you a lot, Surreptitious, for your answer. I think the same at the rate of 80%.Surreptitious57 wrote: A prime number has to be divisible by I and itself not just itself which is the case for I
But I have also heard that it is simply a convention of mathematics that I is not prime
So we do have the same definition.
Except for interpretation; so let us vanish this part saying that 1 cannot be itself its divider. (Or the [by 1 and itself] should be replaced by [by 1 Or itself]  in accordance to logic(s). (I dislike the s for logics, because if it were correct, one were sufficient  and it may mean something like "lyrics", but lyrics vanish themselves, (it is famous)  so please let me write logic, the determinant letting us understand that it is about a substantive).
In fact,
my first/practical worry with all this, is that computers  based on basis Two  have no ponderation number to be prime; in other words: 1, and 0 (okay, about zero).
So howthehell could they compute about algorithms concerned with prime numbers (but not those of encryption as they are timedependant).
The more theoretical problematic I found; actually, conceptual:
If we retrieve 1 from the impair numbers, the algorithms based on impair numbers (as serial finding, or limits) will be complicated. But as the prime numbers do nowadays not follow a "law" (about regularity nor frequency), such that the algorithms founded on them should be highly complicated  (also in case of 1 retrieved).
A mathematicianlambda will think "It as for the impair numbers: that's only 1"  but it does not seem to occur that 1 sounded not to be to extract, and moreover not for a socalled convention.
What do you think about this "convention"?
Depending on me: I have a rule (simply the definition, then the practical skepticism about computers abilities  very bizarre), and the mathematicians have a convention; so that these last shall abandon their mentioned coutume, as there is not a sole reason for to maintain it.
Re: Why is 1 not classified as being a prime number ?
If for no other reason, it's so that the theorem on unique prime power decomposition works out. Otherwise we'd have to say that 15 = 3 x 5 = 1 x 3 x 5 = 1 x 1 x 3 x 5 = 1 x 1 x 1 x 3 x 5 = ...
We would no longer have unique factorization into a product of prime powers. We'd have to say, "Every positive whole number can be uniquely factored into a product of prime powers, not counting those pesky 1's."
To avoid talking about the pesky 1's, we just disallow 1 as a prime.
There's a deeper reason, which is that 1 is a unit in the ring of integers. That means it has a multiplicative inverse, namely itself. It doesn't make sense for any unit to be considered a prime, since any invertible element divides ANYTHING.
To see this more starkly, consider the real numbers, in which every nonzero element has a multiplicative inverse. It doesn't make sense to ask if 1/2 is a prime, since 1/2 divides EVERY real number. So does 2. So does pi. Whenever invertible elements are concerned, it makes no sense to ask if they're prime, because they divide everything and can be divided by everything nonzero.
So there are two good reasons, one practical and one theoretical. The practical reason is to save some verbiage in the statement of the prime power decomposition theorem. The theoretical reason is that 1 is a unit in the ring of integers.
We would no longer have unique factorization into a product of prime powers. We'd have to say, "Every positive whole number can be uniquely factored into a product of prime powers, not counting those pesky 1's."
To avoid talking about the pesky 1's, we just disallow 1 as a prime.
There's a deeper reason, which is that 1 is a unit in the ring of integers. That means it has a multiplicative inverse, namely itself. It doesn't make sense for any unit to be considered a prime, since any invertible element divides ANYTHING.
To see this more starkly, consider the real numbers, in which every nonzero element has a multiplicative inverse. It doesn't make sense to ask if 1/2 is a prime, since 1/2 divides EVERY real number. So does 2. So does pi. Whenever invertible elements are concerned, it makes no sense to ask if they're prime, because they divide everything and can be divided by everything nonzero.
So there are two good reasons, one practical and one theoretical. The practical reason is to save some verbiage in the statement of the prime power decomposition theorem. The theoretical reason is that 1 is a unit in the ring of integers.
Re: Why is 1 not classified as being a prime number ?
Who cares? It doesn't matter!
Re: Why is 1 not classified as being a prime number ?
Can't say I agree. It's much closer to a Who Cares. If 1 were declared prime, we'd just add "ignoring those pesky 1's" to the statement of the theorem on prime power decomposition, and everything else in math would stay the same. This is not in any way a question of blue sky research. It's a triviality. Like asking if 0 is a natural number, another definitional triviality that people like to argue about. Doesn't make the slightest difference to anything, it's just a convention that you can take either way according to your preference.
It's true that 1 being a unit in the ring of integers is a fundamental theoretical reason why 1's not considered a prime. But still, there's just no research component or any relation to cryptography. It's not like someone asked, "Who cares about the Riemann hypothesis." In that case, the remarks about cryptography and blue sky research would be on point.
Re: Why is 1 not classified as being a prime number ?
My point was not about the issue of "1" but a broader observation about the study of primes generally, which was how I interpreted the post I responded to.
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