Post Ai5aOmsyDjxUe2OPc8 by aadmaa@mathstodon.xyz
(DIR) More posts by aadmaa@mathstodon.xyz
(DIR) Post #Ai5ZLKkqFsGQZhNcYK by futurebird@sauropods.win
2024-05-20T17:04:38Z
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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?
(DIR) Post #Ai5a3uAvvFrRAjpyK0 by kellyromanych@mastodon.social
2024-05-20T17:12:11Z
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@futurebird more of a music than math explanation...1 isn't prime because it's the loneliest number 🎶
(DIR) Post #Ai5aMn7COdf4mG61Zo by ColesStreetPothole@weatherishappening.network
2024-05-20T17:16:02Z
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@futurebird Because it is defined as such. 😁 What can I say, sometimes I'm shallow, like the water at a gentle waterfall's edge.
(DIR) Post #Ai5aOmsyDjxUe2OPc8 by aadmaa@mathstodon.xyz
2024-05-20T17:16:15Z
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@futurebird I feel like the termination explanation is kind of fundamental. When you factor a number into its primes, what happens if you allow 1?Then factor, say 6:6 = 3 * 26 = 3 * 2 * 16 = 3 * 2 * 1 * 16 = 3 * 2 * 1 * 1 * 1So that's demented.
(DIR) Post #Ai5bKQyBrRk5TCK2wi by tlariv@mastodon.cloud
2024-05-20T17:26:53Z
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@futurebirdYour 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.
(DIR) Post #Ai5c4fPEWIIjZ116zw by pdcawley@mendeddrum.org
2024-05-20T17:35:05Z
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@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”
(DIR) Post #Ai5cVS1EOv2TrZYtgu by janbogar@mastodonczech.cz
2024-05-20T17:40:04Z
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@futurebird Maybe because then prime factorisation wouldn't be unique. 3x2x1 and 3x2x1x1x1 are both factorisations of 6.
(DIR) Post #Ai5eIABVb3jEh9A1fk by weaselx86@mastodon.social
2024-05-20T17:59:58Z
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@futurebird 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."https://en.wikipedia.org/wiki/Prime_number#Primality_of_one
(DIR) Post #Ai5eUCwRAkWkiir2OG by futurebird@sauropods.win
2024-05-20T18:00:11Z
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@MissConstrue @esther 😳
(DIR) Post #Ai5epAs83tWCNopxA0 by futurebird@sauropods.win
2024-05-20T18:05:22Z
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@weaselx86 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.
(DIR) Post #Ai5eszNqiX7PuXUcMq by barrygoldman1@sauropods.win
2024-05-20T18:05:34Z
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@futurebird if u made a program to factor a number into primes and 1 was allowed, how would you decide to terminate it?
(DIR) Post #Ai5ew1Ii2uiIj2XASO by futurebird@sauropods.win
2024-05-20T18:06:23Z
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@barrygoldman1 That was the first reason I gave. But Dismissed as obtuse.
(DIR) Post #Ai5fA9E3NVUdMH5ooq by barrygoldman1@sauropods.win
2024-05-20T18:09:40Z
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@futurebird were u specifically thinking of factoring algorithms or would it break other algorithms?
(DIR) Post #Ai5fEcvuruASeWkG92 by barrygoldman1@sauropods.win
2024-05-20T18:10:32Z
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@futurebird anyway i subscribe to the reason being that if 1 were prime, nubmers wouldnt have unique prime factorizations
(DIR) Post #Ai5gEAdQBGm2qDRN6u by degreesOfFreedom@denton.social
2024-05-20T18:21:48Z
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@futurebird @weaselx86 For a seriously in-depth treatment of the primality of one, check out https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell1/cald5.pdfandhttps://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html
(DIR) Post #Ai5gu9UunbcvTcTpjs by llewelly@sauropods.win
2024-05-20T18:28:31Z
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@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.
(DIR) Post #Ai5iQujaoFBpCRX0ls by jvluso@towns.gay
2024-05-20T18:46:26Z
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@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.
(DIR) Post #Ai5oUZTGPKEU8U1LQ8 by lufthans@mastodon.social
2024-05-20T19:53:50Z
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@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 harmonicsMath 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 :)
(DIR) Post #Ai5tzemxBc9zNMsAYC by j3b@mastodon.sdf.org
2024-05-20T20:55:57Z
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@futurebird wait what?
(DIR) Post #Ai60UsiQ3X5zGt1YLw by justafrog@mstdn.social
2024-05-20T22:08:55Z
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@futurebird If 1 is a prime, everything is divisible by it, thus defeating the whole point of having prime numbers.The utility is what convinces me.
(DIR) Post #Ai659jVtZbIAQBtviy by soaproot@sfba.social
2024-05-20T23:01:08Z
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@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"
(DIR) Post #Ai6B7c4sqb4YVQdg1I by sabik@rants.au
2024-05-21T00:07:57Z
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@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
(DIR) Post #Ai6I2gaDMuZqi5tK8e by jes5199@mastodon.social
2024-05-21T01:25:31Z
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@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!
(DIR) Post #Ai6uHDvE0bAUQB4tM0 by CliftonR@wandering.shop
2024-05-21T08:33:52Z
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@futurebird @weaselx86 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.
(DIR) Post #Ai6vJ7FNtdqoFjyQCW by IngaLovinde@embracing.space
2024-05-21T08:45:28Z
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@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.
(DIR) Post #Ai7HIk0OVU1HA4Bk7k by jbqueru@fosstodon.org
2024-05-21T12:51:51Z
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@futurebird It's invertible, and primality is defined modulo invertible numbers.
(DIR) Post #Ai7P25oGz5QZUt5iOu by jbqueru@fosstodon.org
2024-05-21T12:56:09Z
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@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))
(DIR) Post #Ai7iCFHkVYSYuSQv6u by mavu@mastodon.social
2024-05-21T17:52:36Z
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@futurebird it's not divisible by itself *and* one.Because itself *is* one.Not perfect, but works for my brain.
(DIR) Post #Ai7pDx8vdAeMtu4JqS by andymandias@mastodon.social
2024-05-21T19:12:00Z
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@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.