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Is 1 Prime, and Does It Matter?

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Re: Is 1 Prime, and Does It Matter?

#111

Post by pyfon »

wesselbindt wrote: post #4

If 1 is prime, then the fundamental theorem of arithmetic goes from "every positive integer can be written as a product* of primes in one and only one way" to "every positive integer can be written as a product of primes greater than 1 in one and only one way". Doesn't quite have the same ring to it. So just from an aesthetic perspective, no I'd rather 1 isn't a prime number. * empty products being 1 of course

That is the first time I thought of 1 as being the product of []

Thay is enough justification for me of 1 not being prime. It has a factorisation!

Re: Is 1 Prime, and Does It Matter?

#113

Post by dullcrisp »

JadeNB wrote: post #20
JJMcJ wrote: post #3

One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1".

> One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1". This is true and compelling as things developed, but I think it's an explanation of where history brought us, rather than a logical inevitability. For example, I can easily imagine, in a different universe, teachers patiently e…

And if we treat zero as not a number, it would make division much easier to define. I wrote that sentence as a joke but now I wonder if maybe it’s true. Does addition really need to have an identity? Maybe we just saw that multiplication has an identity and got a bit carried away. I’m not too sure about this negative number business while we’re at it. Could be that we just took a wrong turn somewhere.

Re: Is 1 Prime, and Does It Matter?

#114

Post by JadeNB »

dullcrisp wrote: post #113
JadeNB wrote: post #20

Earlier quoted context omitted.

> One reason that 1 is often excluded from the prime numbers is that if it was included, it would complicate the theorems, proofs, and exposition by the endless repetition of "not equal to 1". This is true and compelling as things developed, but I think it's an explanation of where history brought us, rather than a logical inevitability. For example, I can easily imagine, in a different universe, teachers patiently e…

And if we treat zero as not a number, it would make division much easier to define. I wrote that sentence as a joke but now I wonder if maybe it’s true. Does addition really need to have an identity? Maybe we just saw that multiplication has an identity and got a bit carried away. I’m not too sure about this negative number business while we’re at it. Could be that we just took a wrong turn somewhere.

> And if we treat zero as not a number, it would make division much easier to define. I wrote that sentence as a joke but now I wonder if maybe it’s true. Does addition really need to have an identity?

It probably doesn't, but, if you want to allow negative numbers, then addition is partial unless you have 0. It's perfectly reasonable to disallow negative numbers—historically, negative numbers had to be explicitly allowed, not explicitly disallowed—but it does mean that subtraction becomes a partial operation or, phrased equivalently but perhaps more compellingly, that we have to give up on solving simple equations for x like x + 2 = 1.

Re: Is 1 Prime, and Does It Matter?

#115

Post by JadeNB »

chongli wrote: post #95
JadeNB wrote: post #62

Earlier quoted context omitted.

> Many (most?) results are easier to write if you allow the empty set. For example: > "The intersection of two sets is a set." Many results in set theory, yes! (Or at least in elementary set theory. I'm not a set theorist by profession, so I can't speak to how often it arises in research-level set theory.) But, once one leaves set theory, the empty set can cause problems. For the first example that springs to mind, i…

I don’t see why it’s a problem that the empty set cannot be a group. The empty set, being empty, lacks an identity element. Thus all groups are non-empty. The same is true for any structure which posits the existence of some element. Of course it cannot be the empty set.

> I don’t see why it’s a problem that the empty set cannot be a group. The empty set, being empty, lacks an identity element. Thus all groups are non-empty.

It's not necessarily a problem that the empty set cannot be a group. (Although the only reason that it cannot is a definition, and, similarly, the definition of a field requires two distinct elements, which hasn't stopped some people from positing that it is a problem that there is then no field with one element.)

The problem is that there's a natural property of magmas (sets with binary operation), namely the uniquely solvability condition I mentioned, that characterizes "group or the empty set," which is more awkward than just characterizing groups. Or you may argue, fairly, that that's not a problem, but it is certainly an example where allowing the empty set to be a set complicates statements, which is all that I was meaning to illustrate. Hopefully obviously, without meaning seriously to suggest that the empty set shouldn't be a set.

(I remembered in the course of drafting this comment that https://golem.ph.utexas.edu/category/2020/08/the_group_with_... discusses, far more entertainingly and insightfully than I do, the characterization that I mention, and may have been where I learned it.)

Re: Is 1 Prime, and Does It Matter?

#116

Post by sepidy »

I think 1 is so different from other numbers, it seems that in the past, some people did consider 1 to be a prime number. However, by the early 1900s, mathematicians agreed to exclude 1 from the list of primes to keep mathematical rules clear and consistent.

Re: Is 1 Prime, and Does It Matter?

#117

Post by samf »

dullcrisp wrote: post #108
samf wrote: post #65

Earlier quoted context omitted.

Some examples are in these comments, e.g. the Fundamental Theorem of Arithmetic. The Sieve of Eratosthenes is an amusing outcome, where 1 is the only prime if you take it literally. But also mentioned elsewhere in the thread: if we declared 1 to be a prime, then many (I daresay "most") of our theorems would have to change "prime number" to "prime number greater than one".

If you defined 1 to be a prime but to not be odd then some theorems could stay the same.

Ha, yes, I was thinking of the theorems that refer to "odd prime" to exclude 2. :-)

Re: Is 1 Prime, and Does It Matter?

#119

Post by margalabargala »

munchler wrote: post #109
margalabargala wrote: post #99

Earlier quoted context omitted.

> 1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be. The other ones, sure, but I'm not following this one.

https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B...

Well that's really fun! I had no idea, thank you.

Re: Is 1 Prime, and Does It Matter?

#120

Post by NoahZuniga »

scoofy wrote: post #92

Just a note from your friendly philosophy degree holder: Axioms are arbitrary. Use the axioms that are the most useful.

While axioms are in some sense arbitrary, it is helpful if they are consistent (informally: you can't prove something that "is false"; formally: you can't prove p and not p). Also other people like it if your axioms feel obvious.
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