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Is 1 Prime, and Does It Matter?

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Re: Is 1 Prime, and Does It Matter?

#61

Post by john-h-k »

Maxatar wrote: post #38
gerdesj wrote: post #27

Earlier quoted context omitted.

I'm no expert but: "...ignoring the trivial case of 1 being an obvious factor of every integer." I remember quite a big chunk of GEB formally defining how integers are really not trivial! The main problem seems to be is that you soon end up with circular reasoning if you are not razor sharp with your definitions. That's just in an explainer book 8) Then you have to define what factor means ...

Correct, it's impossible to specifically and formally define the natural numbers so that addition and multiplication work. Any definition of the natural numbers will also define things that look very similar to natural numbers but are not actually natural numbers.

Can you elaborate on this?

My understanding is you can specifically and formally define the natural numbers with addition and multiplication, although multiplication means the language is no longer decidable.

You can define natural numbers with just addition ( Presburger arithmetic ) and it’s decidable.

Im not sure how undecidable “will define things that are similar to natural numbers but are not” but maybe I am missing something

Re: Is 1 Prime, and Does It Matter?

#62

Post by JadeNB »

gus_massa wrote: post #36
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…

Many (most?) results are easier to write if you allow the empty set. For example: " The intersection of two sets is a set. "

> 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, it is a cute result that, if a set S has a binary operation * such that, for every pair of elements a, b in S, there is a unique solution x to a*x = b, and a unique solution y to y*a = b, then * makes S a group ... unless S is empty!

In fact, on second thought, even in set theory, there are things like: the definition of a partial order being a well ordering would become simpler to state if the empty set were disallowed; and the axiom of choice would become just the statement that the product of sets is a set! I'm sure that I could come up with more examples where allowing empty sets complicates things, just as you could come up with more examples where it simplifies them. That there is no unambiguous answer one direction or the other is why I believe this alternate universe could exist, but we're not in it!

Re: Is 1 Prime, and Does It Matter?

#63

Post by brennopost »

jordigh wrote: post #50
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".

To be fair, 2 is also a very odd prime because it's even. So many theorems have to say, "for every odd prime..." https://math.stackexchange.com/questions/1177104/what-is-an-...

It's actually the least odd prime

Re: Is 1 Prime, and Does It Matter?

#64

Post by [deleted] »

Maxatar wrote: post #55
overboard2 wrote: post #47

Earlier quoted context omitted.

What do you mean by "not actually"? Edit: do you mean literally impossible?

I mean it's logically impossible to formally and specifically define the natural numbers without introducing a logical inconsistency. The best you can do is define a set that has all the properties of natural numbers but will also define things that aren't natural numbers as well. As an analogy you could imagine trying to define the set of all animals with a bunch of rules... "1. Animals have DNA, 2. Animals ingest o…

[deleted]

Re: Is 1 Prime, and Does It Matter?

#65

Post by samf »

rtkwe wrote: post #46
feoren wrote: post #39

All models are wrong, but some models are useful. It's not useful to consider 1 prime, so we don't. You're free to invent a new model of math where 1 is prime and see where it takes you; nobody will be offended. This happens all the time: "but what if we could take the square root of a negative number? What then?", etc. 99% of the time, this leads to a theory that is provably inconsistent and therefore useless. Out o…

I've always wondered what actually breaks if 1 is prime or conversely what defining 1 as not prime gives us. Got just far enough into my math degree before switching to CompSci to stay of of universities the rest of my life to want to know.

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".

Re: Is 1 Prime, and Does It Matter?

#66

Post by alganet »

mikepurvis wrote: post #54
alganet wrote: post #41

"Only divisible by itself and 1" is a darn elegant definition. 1, 2 and 3 are kind of special to me. In prime distribution studies, I discovered that they are special. It gets easier for some things if you consider primes only higher or equal to 5. Explaining distribution gets easier, some proofs become more obvious if you do that (tiny example: draw a ulam-like spiral around the numbers of an analog clock. 2 and 3 w…

The 1 exception matters as well for prime mutuality, like X and Y share no common factors other than 1 of course, sigh .

I see 1 as mostly an anchor. However, my thing is not about working out axioms and formal mathematics. I do some visualizations that can help demonstrate aspects of prime distribution.

I am fascinated by geometric proofs though. The clock thing is just a riff on Ulam's work. I believe there is more to it if one sees it as a geometric object and not just a visualization drawing. I could be wrong though.

Re: Is 1 Prime, and Does It Matter?

#67

Post by meroes »

munchler wrote: post #18

Other good nerd-sniping math questions: 0^0 = 1? Yes, it’s simpler that way. 0! = 1? Yes, it’s simpler that way. 0/0 = ∞? No, it’s undefined. 0.9999… = 1? Yes, it’s just two ways of expressing the same number. 1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be.

0^0 got Gemini 2.5 pro the other day for me. It claimed all indeterminate forms (in the context of limits) are also undefined as a response to a prompt dividing by zero. 0^0 is the most obvious exception, it's typically defined as =1 as you said.

Re: Is 1 Prime, and Does It Matter?

#68

Post by feoren »

rtkwe wrote: post #46
feoren wrote: post #39

All models are wrong, but some models are useful. It's not useful to consider 1 prime, so we don't. You're free to invent a new model of math where 1 is prime and see where it takes you; nobody will be offended. This happens all the time: "but what if we could take the square root of a negative number? What then?", etc. 99% of the time, this leads to a theory that is provably inconsistent and therefore useless. Out o…

I've always wondered what actually breaks if 1 is prime or conversely what defining 1 as not prime gives us. Got just far enough into my math degree before switching to CompSci to stay of of universities the rest of my life to want to know.

The biggest problem is that you lose unique prime factorization. With prime factorization, I get a unique representation of every positive integer. Let's consider a way to write positive integers in "base prime", similar to base 10 or base 2. I'll start counting from 1 and write numbers as a tuple of prime factors. Similar to base 10, "base prime" has an infinite set of 0s that we're leaving out for brevity (e.g. 19 = 0000019), although it's on the right side instead of the left.

    1 = () = (0, 0, 0, 0, 0, ...)
    2 = (1) = (1, 0, 0, 0, 0, ...)
    3 = (0, 1)
    4 = (2)
    5 = (0, 0, 1)
    6 = (1, 1)
    7 = (0, 0, 0, 1)
    8 = (3)
    9 = (0, 2)
    10 = (1, 0, 1)
The i th position in every tuple is the power of the i th prime in the factorization of that number. So 10 = (1, 0, 1) = 2^1 * 3^0 * 5^1. 84 would be (2, 1, 0, 1) = 2^2 * 3^1 * 5^0 * 7^1. If we have unique factorization, there is exactly one way to write every positive integer like this, and there are many insights we can gain from this factorization. If 1 is prime, then we can write 6 = 1^257 * 2^1 * 3^1, or any other power of 1 we like. We just gain nothing from it.

There are often many equivalent ways to define any mathematical object, and I'm sure there are plenty of ways to define a prime number other than "its only factors are itself and 1". These other definitions are likely to obviously exclude 1. One obvious one is the set of basis coordinates in this "unique factorization" space that I just laid out here. And we're never really excluding or making a special case for 1, because 1's factorization is simply the absence of any powers -- empty set, all 0s, whatever you want to call it.

Keep in mind that "unique factorization" turns out to be very interesting in all sorts of other mathematical objects: rings, polynomials, symmetries, vector spaces, etc. They often have their own notion of "prime" or "primitive" objects and the correspondence with integer-primes is much cleaner if we don't consider 1 prime.

Re: Is 1 Prime, and Does It Matter?

#70

Post by samf »

feoren wrote: post #39

All models are wrong, but some models are useful. It's not useful to consider 1 prime, so we don't. You're free to invent a new model of math where 1 is prime and see where it takes you; nobody will be offended. This happens all the time: "but what if we could take the square root of a negative number? What then?", etc. 99% of the time, this leads to a theory that is provably inconsistent and therefore useless. Out o…

This is the best answer.

We could declare 4 to be a prime number, and keep the rest of the definition the same. Instead of just saying "no", you could ask, "okay, what would that do for us?" If there isn't a good answer, then what's the point? And usually, you're not in the 1% of 1% of 1%.

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