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
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.
@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))
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