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Is 1 Prime, and Does It Matter?

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Re: Is 1 Prime, and Does It Matter?

#71

Post by wesselbindt »

tshaddox wrote: post #23
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

It seems a little inconvenient to require acceptance that empty products equal 1, since that is also slightly subtle and deserving of its own explanation of mathematical terminology. Of course, I generally hear the fundamental theorem of arithmetic phrased as “every integer greater than one…” which is making its own little special case for the number 1.

Any convention comes with the inconvenience of definition and explanation. So to call the convention that the empty product equals 1 based on that alone seems a bit unfair. The reason the mathematical community has adopted this convention is because it makes a lot of proofs and theorems a bit easier to state. So yes, you lose a bit of convenience in one spot, and gain a bit in a whole bunch of spots.

And note that this convention is not at all required for the point I'm making regarding prime numbers. As you say yourself, restrict the theorem to integers greater than 1, and you can forget about empty products (and it is still easier to state if 1 is not prime (which it isn't)).

Re: Is 1 Prime, and Does It Matter?

#72

Post by [deleted] »

rvba wrote: post #58
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.

If we try to define division by zero, shouldnt 0/0 be 1? Or even more abstract "every element on y". Which I think could sort of work

[deleted]

Re: Is 1 Prime, and Does It Matter?

#73

Post by gerdesj »

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…

It seems you know what you are on about! Thank you for a cracking comment.

I've always had this feeling that the foundations (integers etc) are a bit dodgy in formal Maths but just as with say Civil Engineering, your world hasn't fallen apart for at least some days and it works. Famously, in Physics involving quantum: "Shut up and calculate".

Thankfully, in the real world I just have to make web pages, file shares and glittery unicorns available to the computers belonging to paying customers. Securely ...

The foundational aspect equivalent of integers in IT might be DNS. Fuck around with either and you come unstuck rather quickly without realising exactly why until you get suitably rigorous ...

I'm also a networking bod (with some jolly expensive test gear) but that might be compared to pencils and paper for Maths 8)

Re: Is 1 Prime, and Does It Matter?

#75

Post by vikingerik »

ks2048 wrote: post #35

Can we declare 2 composite? Kind of annoying to have an even number in there.

2 being the only even prime isn't really anything fundamentally weird. Every prime is the only divisible-by-that-number prime. 2 has nothing unique about that.

We only notice the case for 2 because our human languages happen to define divisible-by-2 as a word and concept. If our languages called divisible-by-3 "treven" or something like that, we'd think it weird that 3 was the only treven prime.

Re: Is 1 Prime, and Does It Matter?

#76

Post by chrismcb »

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 isn't odd at all! And that I'm being pendantic. But you can't say it is very odd, and then I'm the next sentence day "for every odd prime..."

Re: Is 1 Prime, and Does It Matter?

#77

Post by mathgeek »

murderfs wrote: post #48
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…

A good example of this is the natural numbers. Algebraists usually consider zero to be a natural number because otherwise, it's not a monoid and set theorists want zero because it's the size of the empty set. My number theory textbook defined natural numbers as positive integers, but I'm not entirely sure why.

> My number theory textbook defined natural numbers as positive integers, but I'm not entirely sure why.

Since both the inclusion and exclusion of zero are accepted definitions depending on who’s asking, books usually just pick one or define two sets (commonly denoted as N_0 and N_1). Different topics benefit from using one set over the other, as well as having to deal with division by zero, etc. Number theory tends to exclude zero.

Re: Is 1 Prime, and Does It Matter?

#78

Post by pwdisswordfishz »

> One way in which 1 “quacks” like a prime is the way it accords with Euclid’s Lemma, the principle that asserts that if p is a prime, then whenever the product of two integers is divisible by p, one of the two numbers or both must be divisible by p.

This is debunked by https://ncatlab.org/nlab/show/too+simple+to+be+simple#relati...

Re: Is 1 Prime, and Does It Matter?

#79

Post by teytra »

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…

When I was younger I had a period I often was thinking about prime numbers (before I got old and started thinking about the Roman Empire).

I noticed the same as you, and IIRC the (some?) ancient greeks actually had an idea about 1 as not a number, but the unit that numbers were made of. So in a different class.

2 and 3 are also different, or rather all other primes from 5 and up are neighbours to a multiple of 6, (though not all such neighbours are primes of course).

In base-6 all those primes end in 5 or 1. What is the significance? I don't know. I remember that I started thinking that 2*3=6, maybe the sequence of primes is a result of the intertwining of numbersystems in multiple dimensions or whatever? Then I started thinking about the late republic instead. ;)

Re: Is 1 Prime, and Does It Matter?

#80

Post by alganet »

teytra wrote: post #79
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…

When I was younger I had a period I often was thinking about prime numbers (before I got old and started thinking about the Roman Empire). I noticed the same as you, and IIRC the (some?) ancient greeks actually had an idea about 1 as not a number, but the unit that numbers were made of. So in a different class. 2 and 3 are also different, or rather all other primes from 5 and up are neighbours to a multiple of 6, (th…

If you work not only the primes, but also the modulus function value of each non-prime, things get even more interesting than thinking of base changes! To me, it reveals much more.
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