1 is Not Prime

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jewish_scientist
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1 is Not Prime

Postby jewish_scientist » Thu Aug 18, 2016 9:14 am UTC

This video is about why 1 is not considered a prime number. However, I did not like the explanation that was given. I thought about it and I came up with a definition of prime numbers so that 1 is not prime. Does anyone see any problems with this?

Iff X and Y are integers natural numbers and Y/X = an integer, then X is a divisor of Y.
A prime number is any number that has 2 unique divisors.
Last edited by jewish_scientist on Thu Aug 18, 2016 7:51 pm UTC, edited 1 time in total.

PsiCubed
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Re: 1 is Not Prime

Postby PsiCubed » Thu Aug 18, 2016 9:58 am UTC

You gave one good reason to not consider 1 to be prime, and there are many more.

For example, for any prime number, φ(p)=p-1. Yet this isn't true for the number 1.

The bottom line is that whether we regard 1 as a prime or not is a matter of convention. And it is exactly due to the fact that so many different arguments "conspire" to hint at 1 being non-prime, that makes the convention of "1 isn't prime" useful.
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Re: 1 is Not Prime

Postby Soupspoon » Thu Aug 18, 2016 11:12 am UTC

Not on a connection that I can see Youtube things easily, but the explanation that works best for me is that all (whole, positive, >1*) numbers can be built up as a product of a unique accumulation of primes. Each prime would be just itself (zero other primes), every other number a specific (zero or greater, integer) number of each component prime.

2=21
3=31
4=22
5=51
6=21*31
7=71
8=23
9=32
10=21*51
...

You can also mention 20 on each odd line, etc, as you can include 110 or 130, to each of those, but it is just zero 13s (except when it gets to numbers that have just one thirteen, etc) and absence is the same as explicit power-zero. But include 1 as a prime, and you have infinite non-unique sets featuring a choice of 10, 11 ,...1n,... 1. Which helps no-one, in the parts of the Number Theory field where 'building blocks of primes' is apparently supposed to be a thing... So don't do it! ;)


* Actually, >0 suffices, implicit in "whole, positive". 1=20*30*50*... as a unique 'solution'. Meanwhile, zero would still be excluded (without a multiplication by 0n?) and negatives require -12n+1. I think those actually working with such mathematicophilosophy might be able to say more, but just >0 seems a reasonable expectation to me.
Last edited by Soupspoon on Thu Aug 18, 2016 11:23 am UTC, edited 3 times in total.

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Flumble
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Re: 1 is Not Prime

Postby Flumble » Thu Aug 18, 2016 11:14 am UTC

See also this religious war.

Note that while 1 isn't a prime, 0, i and 1/e are. Otherwise you can't factorize all numbers.


Soupspoon wrote:Not on a con ection that I can see Youtube things easily, but the explanation that works best for me is that all (whole, positive, >1) numbers can be built up as a product of a unique accumulation of primes.

The video uses the argument of the FTA (unique factorization), yes. :wink:

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Re: 1 is Not Prime

Postby Soupspoon » Thu Aug 18, 2016 11:32 am UTC

Flumble wrote:See also this religious war.
Ah, I knew I'd mentioned this elsewhere (this forum or otherwise) recently. I'm at the (current) end of that thread, saying much the same thing.

Note that while 1 isn't a prime, 0, i and 1/e are. Otherwise you can't factorize all numbers.
Ah, and I just mentioned 0 (and, though not as far as i, its square) and having added a footnote whilst correcting a typo I'm going to have to rethink the footnote I just added... Or leave it to wiser minds than me.

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Re: 1 is Not Prime

Postby Demki » Thu Aug 18, 2016 11:43 am UTC

Flumble wrote:Note that while 1 isn't a prime, 0, i and 1/e are. Otherwise you can't factorize all numbers.

1/e prime? In what context?

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Re: 1 is Not Prime

Postby Flumble » Thu Aug 18, 2016 11:57 am UTC

Demki wrote:
Flumble wrote:Note that while 1 isn't a prime, 0, i and 1/e are. Otherwise you can't factorize all numbers.

1/e prime? In what context?

Non-integers of course! Otherwise, how would you factorize 1/2 or π? 1/2 is simply (1/e)^{a lot}*{a lot of primes}.

Okay, so maybe something breaks if you have 1=2*(1/e)^{a lot}*{a lot of primes}. But that just means that "unique factorization" has to be restricted to "no strict subset of a factorization may produce 1". :roll:

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Re: 1 is Not Prime

Postby Cleverbeans » Thu Aug 18, 2016 12:42 pm UTC

I assumed this was just by definition to make unique factorization possible. I think of 1 as being conveniently not prime rather than inherently not prime.
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Re: 1 is Not Prime

Postby Zohar » Thu Aug 18, 2016 1:04 pm UTC

1 not being a prime is purely a matter of definition. And what you wrote in the OP, jewish_scientist, is just as valid as saying "A prime number is a natural number whose only divisor is 1 and itself, and is not 1". I mean sure, it's fine as a definition, but it still treats 1 as a special case, just without explicitly mentioning it.

Also, you should change "integer" to "natural number", since 3 is divisible by 3, 1, -1, and -3, and all are integers.
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Demki
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Re: 1 is Not Prime

Postby Demki » Thu Aug 18, 2016 1:09 pm UTC

Flumble wrote:
Demki wrote:
Flumble wrote:Note that while 1 isn't a prime, 0, i and 1/e are. Otherwise you can't factorize all numbers.

1/e prime? In what context?

Non-integers of course! Otherwise, how would you factorize 1/2 or π? 1/2 is simply (1/e)^{a lot}*{a lot of primes}.

Okay, so maybe something breaks if you have 1=2*(1/e)^{a lot}*{a lot of primes}. But that just means that "unique factorization" has to be restricted to "no strict subset of a factorization may produce 1". :roll:

Unless you allow irrational exponents, (1/e)^(n)*m where n and m are nonzero integers is never 1/2, since (1/e)^(n) is irrational for all nonzero integers n.
And if you do allow irrational exponents, you can write any positive integer as (1/e)^x and as 2^(y) with x and y real, and this breaks your 'unique' factorization.

Unless what you mean by 'a lot' is using sequences of irrational numbers to approximate rationals, which is weird to say the least.

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Re: 1 is Not Prime

Postby PsiCubed » Thu Aug 18, 2016 1:42 pm UTC

Flumble wrote:Okay, so maybe something breaks if you have 1=2*(1/e)^{a lot}*{a lot of primes}. But that just means that "unique factorization" has to be restricted to "no strict subset of a factorization may produce 1". :roll:


Defining "prime" over the reals (or the rationals) is problematic - to say the least. Since you can write ANY real number (except 0) as a product of other real numbers, the notion of primility doesn't make much sense in this case.

There's actually an entire mathematical theory dealing with what the words like "prime" might mean in groups other than the integers: Ring theory.

Unique Factorization doesn't always hold. For example, if your "world" is all the numbers of the form a+b*sqrt(5)*i, then:

6 = 2x3 = (1-sqrt(5)*i)x(1+sqrt(5)*i)

with all the above numbers (except the 6) being "prime" in the sense that they cannot be factored in the same "world".
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Re: 1 is Not Prime

Postby Xanthir » Thu Aug 18, 2016 10:40 pm UTC

Zohar wrote:1 not being a prime is purely a matter of definition. And what you wrote in the OP, jewish_scientist, is just as valid as saying "A prime number is a natural number whose only divisor is 1 and itself, and is not 1". I mean sure, it's fine as a definition, but it still treats 1 as a special case, just without explicitly mentioning it.

Yes, this is the important part that people seem to implicitly forget when they ask questions like this. "Prime" isn't a natural quality that we can just recognize in the universe, it's a definition we created to label a particular subset of the integers that have some useful properties. And it so happens that, for most purposes, 1 isn't a useful integer to include in that subset. It also happens that many of the simplest ways to define the subset happen to include 1 as a degenerate case (with that degeneracy being precisely why 1 isn't useful to include), and so we often just exclude 1 explicitly in the definition, rather than trying to craft a definition with tricky wording that excludes the degenerate case. (This is quite normal in all walks of life - definitions often technically include some uninteresting degenerate cases that we all just tacitly ignore, but we have to be a little more precise in math. I write technical standards for a living, and while I enjoy it when I can craft a clever definition that excludes degenerate cases implicitly, I'm quite happy to produce prosaic, easy-to-read definitions that explicitly knock out some uninteresting cases instead.)
Last edited by Xanthir on Thu Oct 06, 2016 6:39 pm UTC, edited 1 time in total.
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Re: 1 is Not Prime

Postby jewish_scientist » Thu Oct 06, 2016 6:01 pm UTC

Have have a new question and I did not want to make a whole new thread because I expect the answer to this question to be rather short. What is the Fundamental Theorem of Arithmetic used for? I understand that it must be a very basic theorem that almost all of mathematics is based on, but at some point in time it had to have been new. When it was first discovered/ invented what proofs used it directly, and what problem lead to its discovery/ invention? Also, I can see how this can be very useful in Number Theory because it says something fundamental about composite and prime numbers. What I cannot see is how it is associated with arithmetic, unless that term refers to more than the study of operations.

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Re: 1 is Not Prime

Postby Nyktos » Thu Oct 06, 2016 9:02 pm UTC

Historically number theory was often referred to as "arithmetic", and occasionally it still is. (This is basically analogous to the way mathematicians use "algebra" and "geometry" to describe branches of mathematics that in their modern incarnations don't have much obvious connection to what non-mathematicians would think of on hearing those terms.)

The theorem goes at least as far back as ancient Greece; Euclid's Elements contains a proof of it. The name "fundamental theorem of arithmetic" is surely much newer but I'm not sure who came up with it. It's hard to come up with specific uses of it because it's so very fundamental and lurks implicitly in the background whenever one multiplies two integers, but an easy example that I recall: when I took a number theory course, one of the things we had to do on the first assignment is prove that gcd(a, b) * lcm(a, b) = ab, without using the fundamental theorem (because that hadn't been covered in the course yet). Doing so is not especially hard, but requires using your brain. With the fundamental theorem, though, it's extremely easy, since you can just write down what the gcd and lcm must be given factorizations of the numbers.

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Re: 1 is Not Prime

Postby ConMan » Thu Oct 06, 2016 9:39 pm UTC

It's also pretty handy in the proof that there are an infinite number of primes.

Also, having FTA then gives you a reason to look at what it takes for a number system to have unique prime factorisations, and what it might mean for a system to *not* have such a seemingly fundamental properties, and that leads to a bunch of interesting areas.
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Re: 1 is Not Prime

Postby Eebster the Great » Fri Oct 07, 2016 12:51 am UTC

The real question is what makes the "fundamental theorem of algebra" fundamental or a theorem of algebra.

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Re: 1 is Not Prime

Postby EmelinaVollmering » Tue Oct 18, 2016 11:52 am UTC

My teacher once explained the theory. Multiple people have multiple opinions about these things. These are conjecture, rather than valid results.

As 1/1 = 1 and only divisible by it'self and 1(this recur). This is the weakness of mathematics that will apply on infinity points( or numbers in your case) but only deviate at one point. So, you will have a separate definition (as the scientists agree to the issue).

For me, I will 100% agree with you.

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Re: 1 is Not Prime

Postby Xanthir » Wed Oct 19, 2016 12:18 am UTC

None of that made a lick of sense.

I'll repeat myself from before: "Prime" isn't a natural quality that we can just recognize in the universe, it's a definition we created to label a particular subset of the integers that have some useful properties. And it so happens that, for most purposes, 1 isn't a useful integer to include in that subset. It also happens that many of the simplest ways to define the subset happen to include 1 as a degenerate case (with that degeneracy being precisely why 1 isn't useful to include), and so we often just exclude 1 explicitly in the definition, rather than trying to craft a definition with tricky wording that excludes the degenerate case.
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