Comments on: What is the smallest prime?
http://www.metafilter.com/120047/What-is-the-smallest-prime/
Comments on MetaFilter post What is the smallest prime?Tue, 18 Sep 2012 13:47:58 -0800Tue, 18 Sep 2012 13:47:58 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60What is the smallest prime?
http://www.metafilter.com/120047/What-is-the-smallest-prime
<a href="http://arxiv.org/abs/1209.2007">What is the smallest prime?</a> "It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers. To find the first prime, we must also know what the first positive integer is. Surprisingly, with the definitions used at various times throughout history, one was often not the first positive integer (some started with two, and a few with three). In this article, we survey the history of the primality of one, from the ancient Greeks to modern times. We will discuss some of the reasons definitions changed, and provide several examples. We will also discuss the last significant mathematicians to list the number one as prime."post:www.metafilter.com,2012:site.120047Tue, 18 Sep 2012 13:42:13 -0800escabechenumbertheoryprimeprimesmathhistoryhistoryofsciencemathematicsnumbersBy: Philosopher Dirtbike
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572343
Thanks! Stuff like this is why I love the Blue. Also, animal videos.comment:www.metafilter.com,2012:site.120047-4572343Tue, 18 Sep 2012 13:47:58 -0800Philosopher DirtbikeBy: Thorzdad
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572347
Huh? Did 1 get the Pluto treatment and was devalued from being a prime? When I was a kid, we were taught that 1 was a prime. Admittedly, that was soon after the Earth cooled, but...comment:www.metafilter.com,2012:site.120047-4572347Tue, 18 Sep 2012 13:49:30 -0800ThorzdadBy: Cash4Lead
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572348
As the man himself <a href="http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/defVII1.html">put it</a>: <em>A unit is that by virtue of which each of the things that exist is called one.</em>
Mathematical realism FTW!comment:www.metafilter.com,2012:site.120047-4572348Tue, 18 Sep 2012 13:49:39 -0800Cash4LeadBy: muddgirl
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572351
1 being a prime is one of those "teach it one way to kids, teach it another way to college students" facts, like Cristopher Columbus.comment:www.metafilter.com,2012:site.120047-4572351Tue, 18 Sep 2012 13:51:16 -0800muddgirlBy: kaibutsu
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572359
Well, the standard definition says that p is prime if the only divisors of p are 1 and itself.
When we consider p=1, we need that the only divisors of p are 1 and 1; everyone knows that 1 and 1 is 2, and 2 doesn't divide 1. Therefore, 1 isn't prime. (ba dum dum.)
More seriously, my real understanding was that the reason to keep 1 out of the primes club is that it necessitates all kinds of ugly special cases when proving anything. There's already enough "Let p be an odd prime" statements lying around the literature; pretty much every theorem would have to include "Let p be a prime not equal to 1..." It makes a mess of things, and that's enough reason to keep it out of the club...comment:www.metafilter.com,2012:site.120047-4572359Tue, 18 Sep 2012 13:54:16 -0800kaibutsuBy: mrgrimm
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572361
<i>1 being a prime is one of those "teach it one way to kids, teach it another way to college students" facts, like Cristopher Columbus.</i>
I was always taught that 2 was the lowest prime, throughout school. Prime was defined as a number than only had <em>two</em> whole factors, itself and 1. 1 was ruled out by definition of the term. (I don't know if that's "right"--haven't read the PDF yet--but I was <em>never</em> taught that 1 was prime.)comment:www.metafilter.com,2012:site.120047-4572361Tue, 18 Sep 2012 13:54:53 -0800mrgrimmBy: fallingbadgers
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572364
Reading the piece it would seem that any of us who grew up with Hardy will be in the "1 is a prime" camp.
As the paper rightly says, this is about definitions and simplicity not proof.comment:www.metafilter.com,2012:site.120047-4572364Tue, 18 Sep 2012 13:56:28 -0800fallingbadgersBy: jeffburdges
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572367
I bought this <a href="http://www.getdigital.de/products/Two_is_the_oddest_prime/lng/en">Two is the oddest prime</a> shirt because my PhD dad once told that joke. It's an incredibly apt sentiment for a finite group theorists, many other algebraists as well, maybe number theorists too.comment:www.metafilter.com,2012:site.120047-4572367Tue, 18 Sep 2012 13:57:47 -0800jeffburdgesBy: muddgirl
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572370
<i>I was always taught that 2 was the lowest prime, throughout school. </i>
I'm sure some kids don't learn that Colombus discovered America.comment:www.metafilter.com,2012:site.120047-4572370Tue, 18 Sep 2012 13:59:26 -0800muddgirlBy: DU
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572371
<i>my real understanding was that the reason to keep 1 out of the primes club is that it necessitates all kinds of ugly special cases when proving anything.</i>
No, 1 is out of the club because otherwise the Fundamental Theorem of Arithmetic wouldn't be true. That's the one where every number has a unique factorization. 10 = 2x5, for instance. But if 1 is prime, then 10 = 2x5 = 1x2x5 = 1x1x2x5 and on and on. That's no good.comment:www.metafilter.com,2012:site.120047-4572371Tue, 18 Sep 2012 13:59:30 -0800DUBy: Nomyte
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572373
I understand why 0! = 1 is a convenient definition, but I'm amused by 0 choose 0 = 1.comment:www.metafilter.com,2012:site.120047-4572373Tue, 18 Sep 2012 14:00:45 -0800NomyteBy: muddgirl
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572376
<i>No, 1 is out of the club because otherwise the Fundamental Theorem of Arithmetic wouldn't be true.</i>
Yeah, I was an overly-rational kid and I always had a dogmatic problem with prime factorization until I learned that 1 isn't really prime.comment:www.metafilter.com,2012:site.120047-4572376Tue, 18 Sep 2012 14:01:58 -0800muddgirlBy: kmz
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572377
In general 1 isn't considered a prime number because it falls into a third category, different from primes and composites: it's a unit.comment:www.metafilter.com,2012:site.120047-4572377Tue, 18 Sep 2012 14:02:33 -0800kmzBy: Philosopher Dirtbike
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572380
<em>No, 1 is out of the club because otherwise the Fundamental Theorem of Arithmetic wouldn't be true. That's the one where every number has a unique factorization.</em>
Did you read the paper? That's dealt with easily.comment:www.metafilter.com,2012:site.120047-4572380Tue, 18 Sep 2012 14:04:05 -0800Philosopher DirtbikeBy: Egg Shen
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572382
I couldn't bear the thought of banishing 1 from the family of primes. It's already the loneliest number.comment:www.metafilter.com,2012:site.120047-4572382Tue, 18 Sep 2012 14:04:37 -0800Egg ShenBy: kmz
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572383
<i>I understand why 0! = 1 is a convenient definition, but I'm amused by 0 choose 0 = 1.</i>
How many ways do you have to choose nothing? Exactly one way.
And really, n^0 (where n!=0) or 0! being 1 has more to do with 1 being the multiplicative identity than just for convenience. The product of an empty set is the multiplicative identity. The sum of an empty set is the additive identity.comment:www.metafilter.com,2012:site.120047-4572383Tue, 18 Sep 2012 14:04:47 -0800kmzBy: koeselitz
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572393
Everything goes downhill after everyone loses all conception of the meaning of being and decides that one is a number.comment:www.metafilter.com,2012:site.120047-4572393Tue, 18 Sep 2012 14:08:14 -0800koeselitzBy: kmz
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572399
<i>It's already the loneliest number.</i>
But two is just as bad as one.comment:www.metafilter.com,2012:site.120047-4572399Tue, 18 Sep 2012 14:10:30 -0800kmzBy: tigrefacile
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572401
Only about a third of the way through but am already struck by the surprising mathematical truth of Madonna's observation that 'one is such a lonely number'. That was all.comment:www.metafilter.com,2012:site.120047-4572401Tue, 18 Sep 2012 14:10:34 -0800tigrefacileBy: Cash4Lead
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572403
<em>In general 1 isn't considered a prime number because it falls into a third category, different from primes and composites: it's a unit.</em>
But, as the article states, the moderns redefined one as a number, which brings the question of its primeness into view.comment:www.metafilter.com,2012:site.120047-4572403Tue, 18 Sep 2012 14:10:57 -0800Cash4LeadBy: dances_with_sneetches
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572404
So nobody agrees with me that 12 should be considered the lowest prime number?comment:www.metafilter.com,2012:site.120047-4572404Tue, 18 Sep 2012 14:11:07 -0800dances_with_sneetchesBy: tigrefacile
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572408
May Egg Shen be banished to the lonely island of One for scooping me.comment:www.metafilter.com,2012:site.120047-4572408Tue, 18 Sep 2012 14:12:14 -0800tigrefacileBy: MetalFingerz
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572416
This seems like one of those things that make people wonder what academics do all day long.comment:www.metafilter.com,2012:site.120047-4572416Tue, 18 Sep 2012 14:15:30 -0800MetalFingerzBy: Blazecock Pileon
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572418
When a despondent Optimus Prime went on a three-day bender, that was a pretty low moment.comment:www.metafilter.com,2012:site.120047-4572418Tue, 18 Sep 2012 14:16:48 -0800Blazecock PileonBy: Mblue
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572424
1 is first, or the start, or the <em>prime</em> no matter the math, 1 begins.comment:www.metafilter.com,2012:site.120047-4572424Tue, 18 Sep 2012 14:21:08 -0800MblueBy: King Bee
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572448
<a href="http://www.metafilter.com/120047/What-is-the-smallest-prime#4572347">></a> <i>When I was a kid, we were taught that 1 was a prime. </i>
Unless you are centuries old, you had poor math teachers using sloppy definitions.comment:www.metafilter.com,2012:site.120047-4572448Tue, 18 Sep 2012 14:34:58 -0800King BeeBy: King Bee
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572456
<a href="http://www.metafilter.com/120047/What-is-the-smallest-prime#4572424">></a> <i>1 is first, or the start, or the</i> prime <i>no matter the math, 1 begins.</i>
I don't know if maybe you're joking, but this is exactly the problem. "Prime" doesn't mean what you want it to mean, nor does it carry with it any of the meaning it has in normal English usage. In mathematics, we often take over words and say "here's what this means from now on, OK? It doesn't mean what it means in English normally, got it?"
One of my favorite examples is the property that a set of real numbers can have, called <a href="http://en.wikipedia.org/wiki/Nowhere_dense_set">nowhere dense</a>. Again, we've taken over those words. So, what do you call a set of real numbers whose closure does not have empty interior? That's right. <b>Not nowhere dense.</b>comment:www.metafilter.com,2012:site.120047-4572456Tue, 18 Sep 2012 14:38:34 -0800King BeeBy: ceribus peribus
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572467
This is like when they started adding more letters to the list of vowels.comment:www.metafilter.com,2012:site.120047-4572467Tue, 18 Sep 2012 14:44:56 -0800ceribus peribusBy: Egg Shen
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572471
Of course, the <em>largest</em> <small>(known)</small> prime is 2^43,112,609 - 1.
I think of it lurking waaay out there, malignant in its indivisibility, biding its time.comment:www.metafilter.com,2012:site.120047-4572471Tue, 18 Sep 2012 14:46:33 -0800Egg ShenBy: King Bee
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572484
Heh. It's ridiculous to even try and fathom how large that number is. There are only something like 10^80 subatomic particles in the observable universe, right? That Mersenne prime you cite laughs at that number.comment:www.metafilter.com,2012:site.120047-4572484Tue, 18 Sep 2012 14:55:09 -0800King BeeBy: DynamiteToast
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572487
<i>my real understanding was that the reason to keep 1 out of the primes club is that it necessitates all kinds of ugly special cases when proving anything.
No, 1 is out of the club because otherwise the Fundamental Theorem of Arithmetic wouldn't be true. That's the one where every number has a unique factorization. 10 = 2x5, for instance. But if 1 is prime, then 10 = 2x5 = 1x2x5 = 1x1x2x5 and on and on. That's no good.</i>
Not to mention plenty of other formulas, like the one Euclid himself devised for finding <a href="http://en.wikipedia.org/wiki/Perfect_number#Even_perfect_numbers">even perfect numbers</a>.comment:www.metafilter.com,2012:site.120047-4572487Tue, 18 Sep 2012 14:55:56 -0800DynamiteToastBy: kmz
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572493
<i>There are only something like 10^80 subatomic particles in the observable universe, right? That Mersenne prime you cite laughs at that number.</i>
Pshh. <a href="http://en.wikipedia.org/wiki/Graham's_number">Graham's Number</a> doesn't even notice that Mersenne Prime as it casually grinds it into dust.comment:www.metafilter.com,2012:site.120047-4572493Tue, 18 Sep 2012 14:58:54 -0800kmzBy: michaelh
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572498
It's going to be really disappointing when we finally discover the largest prime number that can be expressed without using up all the atoms in the universe. I hope we'll figure out a way to predict primes by then. That or create more atoms/import them from another universe.comment:www.metafilter.com,2012:site.120047-4572498Tue, 18 Sep 2012 14:59:55 -0800michaelhBy: michaelh
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572499
I guess by that point we'll be able to count using sub-sub-subatomic particles. There's hope!comment:www.metafilter.com,2012:site.120047-4572499Tue, 18 Sep 2012 15:00:37 -0800michaelhBy: King Bee
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572506
kmz, one of my colleagues/friends and I often use such numbers to describe how many things we have to do before <i>next Tuesday</i> or whatever the hell. The joke never gets old!
We also laugh sometimes at one of my proofs in graduate school, where I was able to prove an upper bound on some number of 2^(2^40). The real "answer" ended up being 15, so I was close!comment:www.metafilter.com,2012:site.120047-4572506Tue, 18 Sep 2012 15:04:22 -0800King BeeBy: Chuffy
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572538
If you've seen one, you've seen them all?comment:www.metafilter.com,2012:site.120047-4572538Tue, 18 Sep 2012 15:24:14 -0800ChuffyBy: baf
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572568
<em>It's going to be really disappointing when we finally discover the largest prime number that can be expressed without using up all the atoms in the universe.</em>
Dude, no. Internal contradiction there. If there exists a prime number that we can accurately describe as "the largest prime number that can be expressed without using up all the atoms in the universe", then we can express a larger one as follows: "One plus the product of all prime numbers less than or equal to the largest number that can be expressed without using up all the atoms in the universe."comment:www.metafilter.com,2012:site.120047-4572568Tue, 18 Sep 2012 15:36:44 -0800bafBy: baf
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572571
(Or, more simply and certainly: "The smallest prime number greater than TLPNTCBEWUUATAITU".)comment:www.metafilter.com,2012:site.120047-4572571Tue, 18 Sep 2012 15:38:39 -0800bafBy: Obscure Reference
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572578
(That won't necessarily be prime, but will have a larger prime divisor.)comment:www.metafilter.com,2012:site.120047-4572578Tue, 18 Sep 2012 15:39:48 -0800Obscure ReferenceBy: mykescipark
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572591
<em>May Egg Shen be banished to the lonely island of One for scooping me.
posted by tigrefacile at 2:12 PM on September 18 [+] [!]
</em>
I'm pretty sure he was quoting a <a href="http://youtu.be/UiKcd7yPLdU">much earlier source</a>.comment:www.metafilter.com,2012:site.120047-4572591Tue, 18 Sep 2012 15:44:41 -0800mykesciparkBy: mrgrimm
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572617
<i>How many ways do you have to choose nothing? Exactly one way.</i>
1 + (-1), 2 + (-2), 3 + (-3), 4 + (-4), ... ∞ + (-∞)
<a href="http://everythingforever.com/st_math.htm">When Zero = Infinity</a> (God's (!) Math)comment:www.metafilter.com,2012:site.120047-4572617Tue, 18 Sep 2012 15:52:30 -0800mrgrimmBy: JHarris
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572633
Some time ago I responded to a comment in a thread that stated, if there were no life around, there would still be math. My response went along the lines of disagreeing with that statement, but letting it pass because it was quibbling over terms.
But it's still important to quibble over them, and this is why. Even our definition of prime numbers is ad-hoc, based on whatever is most useful to mathematicians. But that's okay. This is because the universe doesn't believe in prime numbers; it has deeper patterns, patterns that we struggle to describe or even notice, of which prime numbers are but a consequence. That is why math is a human thing; it seeks to describe a universe of transcendent complexity. The "true math" is so profound that any creature we could imagine and call "God" would be inadequate to invent it.comment:www.metafilter.com,2012:site.120047-4572633Tue, 18 Sep 2012 15:57:49 -0800JHarrisBy: King Bee
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572647
mrgrimm, that article reads like it was written by a complete hack. (Although, I get it's just some philosopher waxing crazy about nonsense, but still.)
On preview, holy shit, JHarris. That last sentence you wrote is amazing.comment:www.metafilter.com,2012:site.120047-4572647Tue, 18 Sep 2012 16:03:36 -0800King BeeBy: stebulus
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572648
Nomyte: <i>I understand why 0! = 1 is a convenient definition, but I'm amused by 0 choose 0 = 1.</i>
Well, but (0 choose 0) = 0! / 0! 0!, so the one definition strongly suggests the other.
(As does the recurrence (n choose k) = (n-1 choose k) + (n-1 choose k-1), of course.)comment:www.metafilter.com,2012:site.120047-4572648Tue, 18 Sep 2012 16:03:51 -0800stebulusBy: Egg Shen
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572674
<em> Graham's Number doesn't even notice that Mersenne Prime as it casually grinds it into dust.</em>
<a href="http://www.metafilter.com/89927/Very-large-numbers">Put it back in your pants, gentlemen.</a>comment:www.metafilter.com,2012:site.120047-4572674Tue, 18 Sep 2012 16:16:12 -0800Egg ShenBy: baf
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572693
I'm with kmz: "0 choose 0 = 1" makes intuitive sense to me in a way that "0! = 1" doesn't. "0! = 1" is a matter of abstract convenience. "0 choose 0 = 1" is a reflection of the fact that there's only one way to take nothing out of an empty box.comment:www.metafilter.com,2012:site.120047-4572693Tue, 18 Sep 2012 16:25:48 -0800bafBy: King Bee
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572695
n! can be used as the number of ways to arrange n objects in a row. So, 1! is 1, 2! is 2, 3! is 6, and so on. How many ways can you arrange 0 things in row? Just 1, don't do anything.comment:www.metafilter.com,2012:site.120047-4572695Tue, 18 Sep 2012 16:28:31 -0800King BeeBy: jamjam
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572712
But, but there's no way to take nothing out of an empty box, because no matter how much of it you take, it's still all there!comment:www.metafilter.com,2012:site.120047-4572712Tue, 18 Sep 2012 16:36:03 -0800jamjamBy: stebulus
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572735
<i>"0 choose 0 = 1" is a reflection of the fact that there's only one way to take nothing out of an empty box.</i>
The naive intuition for this question is, in my experience, that taking nothing out of an empty box is doing nothing, a non-action, which means there is nothing to do, it cannot be done, so there are zero ways to do it. For most people, it takes some experience with combinatorics (not much, but some) to adjust one's intuition for such scenarios to match what the abstract patterns dictate as the One True Answer, just as it takes a little experience to adjust one's intuition to 0! = 1 (and similar constructions, such as the intersection of no sets, or universal quantification over an empty set), and just as, at a much lower level, it takes some work to get used to the idea that zero is a number and can be reasoned about uniformly with other numbers.
But maybe that wasn't your experience when you studied combinatorics for the first time, in which case I'd like to hear about it.comment:www.metafilter.com,2012:site.120047-4572735Tue, 18 Sep 2012 16:46:43 -0800stebulusBy: jepler
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572738
See <a href="http://en.wikipedia.org/wiki/Empty_product">Empty product</a> for some other arguments that would lead to the conclusion that 0! = 1, since 0! would be equal to any other empty product.
Another line of argument would be to regard factorial as being defined in terms of Γ rather than the other way around (n! = Γ(n-1), n >= 0)comment:www.metafilter.com,2012:site.120047-4572738Tue, 18 Sep 2012 16:47:06 -0800jeplerBy: jepler
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572741
For me, the most interesting part of the original article is the bit where 1 is not a number at all.comment:www.metafilter.com,2012:site.120047-4572741Tue, 18 Sep 2012 16:49:18 -0800jeplerBy: Twang
http://www.metafilter.com/120047/What-is-the-smallest-prime#4572828
The whole question is easily settled. Simply define 1 as a <i>dwarf</i> prime.comment:www.metafilter.com,2012:site.120047-4572828Tue, 18 Sep 2012 17:31:05 -0800TwangBy: leahwrenn
http://www.metafilter.com/120047/What-is-the-smallest-prime#4573061
We spent a not insignificant amount of time in my graduate algebra course discussing whether we were defining 1 as a prime or not. Good times.
<em>One of my favorite examples is the property that a set of real numbers can have, called nowhere dense. Again, we've taken over those words. So, what do you call a set of real numbers whose closure does not have empty interior? That's right. Not nowhere dense.</em>
My students have a real problem with the fact that a set which is not convex is called a...wait for it...non-convex set. (Convex means given any two points in the set, the line segment between the two points is in the set. That is, no pointy-in bits.) They're like "but, isn't it concave?" and I have to explain that no, it's non-convex. (one brought me his little sister's high school math book to show me a picture labeled as a 'concave' set. Still not buying it. )comment:www.metafilter.com,2012:site.120047-4573061Tue, 18 Sep 2012 20:06:28 -0800leahwrennBy: delmoi
http://www.metafilter.com/120047/What-is-the-smallest-prime#4573063
<blockquote><i> One of my favorite examples is the property that a set of real numbers can have, called nowhere dense. Again, we've taken over those words. So, what do you call a set of real numbers whose closure does not have empty interior? That's right. Not nowhere dense.</i></blockquote>
Randomly clicking around on wikipedia the other day I discovered <a href="http://en.wikipedia.org/wiki/Smooth_number">smooth numbers</a>, which actually have to do with their factorization. A number with no prime factors larger then some integer <i>k</i> is a k-smooth number.
This of course leads to <a href="http://en.wikipedia.org/wiki/Smooth_number#Powersmooth_numbers">powersmooth numbers</a>comment:www.metafilter.com,2012:site.120047-4573063Tue, 18 Sep 2012 20:07:34 -0800delmoiBy: stebulus
http://www.metafilter.com/120047/What-is-the-smallest-prime#4573104
<i>They're like "but, isn't it concave?" and I have to explain that no, it's non-convex.</i>
Heh. Just a few months ago I gave a talk where I had to (briefly) mention this. My example was just a set containing two distinct points; it is certainly not convex, but it doesn't seem right to say that the gap between the two points is a concavity.comment:www.metafilter.com,2012:site.120047-4573104Tue, 18 Sep 2012 20:44:51 -0800stebulusBy: lubujackson
http://www.metafilter.com/120047/What-is-the-smallest-prime#4573183
Seems just a semantic difference in how you define prime numbers. It doesn't seem to have much practical use - if there is a practical difference, then you have two equally interesting sets of "prime-like" numbers... but until then, not very interesting.comment:www.metafilter.com,2012:site.120047-4573183Tue, 18 Sep 2012 22:00:23 -0800lubujacksonBy: crazy_yeti
http://www.metafilter.com/120047/What-is-the-smallest-prime#4573252
<i>taking nothing out of an empty box is doing nothing, a non-action, which means there is nothing to do, it cannot be done</i>
Doing nothing is the easiest action of all!comment:www.metafilter.com,2012:site.120047-4573252Tue, 18 Sep 2012 23:21:59 -0800crazy_yetiBy: kaibutsu
http://www.metafilter.com/120047/What-is-the-smallest-prime#4573311
<em>No, 1 is out of the club because otherwise the Fundamental Theorem of Arithmetic wouldn't be true. That's the one where every number has a unique factorization. 10 = 2x5, for instance. But if 1 is prime, then 10 = 2x5 = 1x2x5 = 1x1x2x5 and on and on. That's no good.</em>
Right; and it's immediately fixed by saying that every number has a unique factorization into primes not equal to one. The statement loses some elegance in exchange for expanding the definition of prime number.
<em>Dude, no. Internal contradiction there. If there exists a prime number that we can accurately describe as "the largest prime number that can be expressed without using up all the atoms in the universe", then we can express a larger one as follows: "One plus the product of all prime numbers less than or equal to the largest number that can be expressed without using up all the atoms in the universe."</em>
Another way to get there:
"Erdos said it and I'll say it again,
<a href="http://en.wikipedia.org/wiki/Bertrand%27s_postulate">There's always a prime between n and 2n</a>."comment:www.metafilter.com,2012:site.120047-4573311Wed, 19 Sep 2012 01:47:27 -0800kaibutsuBy: three blind mice
http://www.metafilter.com/120047/What-is-the-smallest-prime#4573326
<i>Only about a third of the way through but am already struck by the surprising mathematical truth of Madonna's observation that 'one is such a lonely number'. That was all.</i>
<a href="http://www.youtube.com/watch?v=Tz79ZYo0erQ">Three Dog Night</a> coined this in 1969.comment:www.metafilter.com,2012:site.120047-4573326Wed, 19 Sep 2012 02:16:50 -0800three blind miceBy: Segundus
http://www.metafilter.com/120047/What-is-the-smallest-prime#4573331
<em>Unless you are centuries old, you had poor math teachers using sloppy definitions.</em>
Words cannot express my feelings when I discovered that my daughter was being taught about the biological category 'mini-beasts' - worms, ants, centipedes, snails...comment:www.metafilter.com,2012:site.120047-4573331Wed, 19 Sep 2012 02:33:24 -0800SegundusBy: stebulus
http://www.metafilter.com/120047/What-is-the-smallest-prime#4574271
<i>Seems just a semantic difference in how you define prime numbers. It doesn't seem to have much practical use</i>
If you deal only with integers, I think you have a point (though I'm not completely sure what you mean by "practical use" in this context). But when you move beyond the integers, and want to understand the analogous phenomena for other sets of numbers (or numberlike objects), it becomes important to straighten out these concepts. As the article says:
<blockquote>The generalization of prime to unique factorization domains clarified the role of unity and now informs the way we define primality in the ordinary positive integers. (p.2)</blockquote>
A similar question is whether the negatives of primes should be considered prime. When dealing with integers only, it seems like just an annoying technical issue of definitional convenience (e.g., it will affect how you state the fundamental theorem of arithmetic, as discussed upthread), but actually it's the shadow of a more serious conceptual issue in a larger, more abstract, context, the resolution of which is to define primeness as a property not of numbers but of ideals.
Mathematicians see this kind of thing a lot, so some of us don't dismiss so-called "semantic" questions quite as readily as people in other fields do.
Another argument for giving thoughtful consideration to such apparently minor definitional issues was given by Dijkstra, who thought it <em>methodologically</em> important to have tidy formalisms, to support rigorous reasoning about large formal objects such as computer programs:
<blockquote>The programmer applies mathematical techniques in an environment with an unprecedented potential for complication; this circumstance makes him methodologically very, very conscious of the steps he takes, the notations he introduces etc. Much more than the average mathematician he is explicitly concerned with the effectiveness of this argument, much more than the average mathematician he is consciously concerned with the mathematical elegance of his argument. He simply has to, if he refuses to be drowned in unmastered complexity. [<a href="http://www.cs.utexas.edu/~EWD/transcriptions/EWD06xx/EWD641.html">EWD641</a>]</blockquote>
<blockquote>To my surprise, and somewhat to my disappointment, I found a lot of mathematicians with respect to their own work not very "elegance-conscious". I found them doing all sorts of clumsy things, clumsy things that I, as a programmer, had already learned to avoid many years ago. [...] The reason why, so often, mathematicians can come away with a rather inelegant way of working is probably that, at least in comparison to software projects, mathematical projects are relatively "small". [<a href="http://www.cs.utexas.edu/~EWD/transcriptions/EWD06xx/EWD619.html">EWD619</a>]</blockquote>To see how this shakes out in mathematical practice, see, for example, Dijkstra's treatment of the infinitude of primes in <a href="http://www.cs.utexas.edu/~EWD/transcriptions/EWD12xx/EWD1203.html">EWD1203</a> and his criticism there of the proof as presented by Kac and Ulam, or his criticism in <a href="http://www.cs.utexas.edu/~EWD/transcriptions/EWD09xx/EWD993.html">EWD993</a> of a statement on prime factorization. His criticisms would strike even many mathematicians as pedantic.comment:www.metafilter.com,2012:site.120047-4574271Wed, 19 Sep 2012 12:35:55 -0800stebulus