What is the best explanation you’ve heard for 1 not being a prime number? For me it’s “because it breaks everything in my programs since the loops won’t terminate” but that’s obtuse. “Because the God of math decrees it so!” is compelling, but shallow.
“it can only be divided by 1 distinct number” is contrived.
1 “feels” prime— it has the fewest factors. (Primeness being about NOT having factors) ruling it out for having too few? eh.
“it’s the zero of multiplication” is better… thoughts?
@futurebird one more way to thing about it: imagine a half-line of points / vectors with non-negative integer coordinates. There is a zero, and there is a "smallest" (cannot be represented as sum of two others) vector, 1.
Now imagine a quarter-plane, there will be two "smallest" vectors besides zero. They're interesting because we can represent any other vector in our quarter-plane as a sum of these "smallest" vectors (and not just sum but an unique sum). Of course we're interested in smallest non-zero vectors, otherwise zero vector would be the only smallest one. What we're interested in are "generating" vectors, those that define a shape of that quarter-plane and its content, and zero vector doesn't define anything.
We can then do the same exercise with 1/8th of 3-dimensional space, etc.
Now extend this to the space with countably many dimensions (and vectors with finite number of non-zero coordinates). And define the mapping between this space and positive integers: vector with a_i coordinate at ith place is converted to the product of ith prime numbers to the a_ith degree. Then vector addition turns into integer multiplication, "smallest" vectors turn into their respective primes, and origin / zero vector is converted to 1.
1 is a prime in the same sense as zero is the smallest vectors, but this doesn't get is anywhere, we're interested in smallest non-zero vectors, those that generate everything else.
@futurebird it breaks the fundamental theorem of arithmetic: that every integer > 1 is a unique product of primes. if 1 was a prime then every integer's factors could include an arbitrary number of 1s, leading to an infinite number of factorizations for each integer.
@esther@futurebird Like, I know all of the words you used, and the sentence reads like English, but damned if I know what any of that means. 🤣. I think I have a psychological block on maths. One of the only “F’s” I got in school was in second grade, and I erased and redid a problem so many times, I’d worn a hole in my paper, so I asked for a new sheet, and the teacher held it up and made fun of me in front of the whole class, saying “Is anyone else having trouble or is it just this dummy?”
From that day forward, I would do enough maths to get past whatever academic hurdle required it, and then drop it like an ex. 🤣
(Found out later, said teacher was one of my father’s brief conquests, as he was a charismatic man with no sense of ethics and a large desire not to wear pants around ladies, and so the teacher bullied me that whole year. Hence the reason I subsequently was educated by nuns. 🤣
@futurebird imho the most compelling reason 1 is not a prime is because it would be terribly inconvenient for many theorems that revolve around primes. The fundamental theorem of arithmetic is what I’m thinking of primarily, but I assume there are others.
But that’s largely because I feel there is too much mystical platonism in the math education I received, eliding the fact that these are systems we humans created for both utilization and beauty.
@futurebird my understanding is that a lot of definitions would have to refer to “primes other than one” and it just saves time to kick one out. unique prime factorizations being the most important. so it is very related to being the zero of multiplication!
@futurebird
Yeah, 1 is neither prime nor composite, it's a secret third thing (a unit)
This is also the reason why prime numbers don't make sense with fractions or with real numbers - all of them are units (except 0), so none are prime or composite
@futurebird There are better answers on this thread for why to pick one definition or another, but as for how to neatly state the definition which includes two and excludes one: "a prime is a natural number which has exactly two distinct factors"
@futurebird heard Queen guitarist and physicist Brian May talk about making the stomping part for We Will Rock You. They had a few people do the stomping and clapping, then he looped them. For the loops he added delays. The delays were prime numbers in order to avoid harmonics
Math is a language we create to describe the universe, so what do we need prime numbers to be for the construct of our language?
I recall prime being itself and 1 as only factors rather than itself and its evil twin :)
@futurebird
if you define prime so that 1 is prime, you get a number system which is equally valid, but so full of anger at having been long snubbed by mathematicians, it plots to overthrow the normal order.
There is probably a history of mathematical papers arguing about whether 1 should be considered to be a prime number...
"In the mid-18th century Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be prime. In the 19th century many mathematicians still considered 1 to be prime, and lists of primes that included 1 continued to be published as recently as 1956."
This is kind of wild since it means that the idea of a prime preceded a well formed definition. Everyone knew what it was in some more general sense before the edges were nailed down.
@futurebird@weaselx86 this is how a lot of math terms are. The set theory definitions of integer addition and subtraction, which form the basis of arithmetic and higher math weren't formally defined until the 1920s, but the concepts of addition and subtraction were widely used and agreed on for thousands of years before that. The definition that gets formalized is the one that's the most useful in the most situations.
The Greeks in the late classic period must have had a basic understanding of primes. The famous Sieve of Eratosthenes method for finding primes by elimination was attributed to Eratosthenes in the 3rd century BCE, and definitely dates to before the 2nd century CE per Wikipedia.
Going back to your original question, I thought about it a while; I think considering 1 to be prime would wreck the valuable concept of all integers > 1 having a unique prime factorization.
that being said hellenic ontology regarding number as ratio makes it hard to engage with 1 as a prime, as it's the base, the unit. 2 is simply twice the unit so the question becomes if any other ratio is 2
4 is more than one ratio it's twice twice but also fouring
there was a more concrete understanding of number in the sense of not abstract relative to unit, but that wasn't used in math it was used in logistic, applied math, trade
egyptian fractions also require some level of dealing with primality because except 3/4 and 2/3, fractions were represented as series of unit fractions. so you needed to be able to do prime expansions
and the use of base 60 in mesopotamian math itself shows an understanding of the utility of primes
when you live in a world having to manually divide the material world, primes are both powerful and a pain
it's hard to really feel concerned about treatment of 1 as a prime before the 1920s and hangings on after because engaging with that using our modern understanding of primality is ahistorical - just like trying to equate hellenic concepts of what math even was with modern sensibilities
classical mathematics, which is usually the foundation people are taught math today is itself something only beginning to be developed in the late 19th and early 20th century
this was a period of recognizing that much of what was taken for granted in math led to quite serious foundational inconsistency: see the foundational crisis of mathematics
we can adhoc together a classical mathematics that takes 1 as prime, but it's simply making another math - it's not equivalent to the historic epistemic understanding of what that would have meant to someone before incompleteness and axiomatic set theory
making a math that defines things in a way that's not convention is absolutely valid (and very cool and fun). you just need to make sure it's well defined and consistent.
@futurebird I quite like: “because mathematicians got sick of writing ‘the set of prime numbers excluding one’ and redefined ‘prime numbers’ so that in far fewer instances they had to take about ‘the union of {1} and the primes’ instead”
@futurebird
Your last answer is the closest to what I think of as the "real" one. One is the multiplicative identity; it's more special than just some prime number.
@winter@glitzersachen@futurebird Yeah, the definition of "only divisible by 1 and itself" is only valid for natural numbers, but gets weird in larger sets. E.g. if you include negative numbers, 2 is still a prime, but it is divisible by 2, -2, 1 and -1.
(and, weirdly, 2 is not a prime in gaussian integers, since it is (1+i)*(1-i))