September 18, 2012 1:42 PM Subscribe

What is the smallest prime? "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."

posted by escabeche (61 comments total) 46 users marked this as a favorite

posted by escabeche (61 comments total) 46 users marked this as a favorite

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...

posted by Thorzdad at 1:49 PM on September 18, 2012 [4 favorites]

posted by Thorzdad at 1:49 PM on September 18, 2012 [4 favorites]

As the man himself put it: *A unit is that by virtue of which each of the things that exist is called one.*

Mathematical realism FTW!

posted by Cash4Lead at 1:49 PM on September 18, 2012 [1 favorite]

Mathematical realism FTW!

posted by Cash4Lead at 1:49 PM on September 18, 2012 [1 favorite]

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.

posted by muddgirl at 1:51 PM on September 18, 2012 [6 favorites]

posted by muddgirl at 1:51 PM on September 18, 2012 [6 favorites]

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...

posted by kaibutsu at 1:54 PM on September 18, 2012 [10 favorites]

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...

posted by kaibutsu at 1:54 PM on September 18, 2012 [10 favorites]

I was always taught that 2 was the lowest prime, throughout school. Prime was defined as a number than only had

posted by mrgrimm at 1:54 PM on September 18, 2012 [1 favorite]

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.

posted by fallingbadgers at 1:56 PM on September 18, 2012 [1 favorite]

As the paper rightly says, this is about definitions and simplicity not proof.

posted by fallingbadgers at 1:56 PM on September 18, 2012 [1 favorite]

I bought this Two is the oddest prime 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.

posted by jeffburdges at 1:57 PM on September 18, 2012 [2 favorites]

posted by jeffburdges at 1:57 PM on September 18, 2012 [2 favorites]

I'm sure some kids don't learn that Colombus discovered America.

posted by muddgirl at 1:59 PM on September 18, 2012 [2 favorites]

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.

posted by DU at 1:59 PM on September 18, 2012 [17 favorites]

I understand why 0! = 1 is a convenient definition, but I'm amused by 0 choose 0 = 1.

posted by Nomyte at 2:00 PM on September 18, 2012

posted by Nomyte at 2:00 PM on September 18, 2012

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.

posted by muddgirl at 2:01 PM on September 18, 2012 [1 favorite]

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.

posted by kmz at 2:02 PM on September 18, 2012

posted by kmz at 2:02 PM on September 18, 2012

Did you read the paper? That's dealt with easily.

posted by Philosopher Dirtbike at 2:04 PM on September 18, 2012 [4 favorites]

I couldn't bear the thought of banishing 1 from the family of primes. It's already the loneliest number.

posted by Egg Shen at 2:04 PM on September 18, 2012 [7 favorites]

posted by Egg Shen at 2:04 PM on September 18, 2012 [7 favorites]

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.

posted by kmz at 2:04 PM on September 18, 2012 [4 favorites]

Everything goes downhill after everyone loses all conception of the meaning of being and decides that one is a number.

posted by koeselitz at 2:08 PM on September 18, 2012 [3 favorites]

posted by koeselitz at 2:08 PM on September 18, 2012 [3 favorites]

But two is just as bad as one.

posted by kmz at 2:10 PM on September 18, 2012 [3 favorites]

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.

posted by tigrefacile at 2:10 PM on September 18, 2012

posted by tigrefacile at 2:10 PM on September 18, 2012

But, as the article states, the moderns redefined one as a number, which brings the question of its primeness into view.

posted by Cash4Lead at 2:10 PM on September 18, 2012

So nobody agrees with me that 12 should be considered the lowest prime number?

posted by dances_with_sneetches at 2:11 PM on September 18, 2012 [12 favorites]

posted by dances_with_sneetches at 2:11 PM on September 18, 2012 [12 favorites]

May Egg Shen be banished to the lonely island of One for scooping me.

posted by tigrefacile at 2:12 PM on September 18, 2012

posted by tigrefacile at 2:12 PM on September 18, 2012

This seems like one of those things that make people wonder what academics do all day long.

posted by MetalFingerz at 2:15 PM on September 18, 2012

posted by MetalFingerz at 2:15 PM on September 18, 2012

When a despondent Optimus Prime went on a three-day bender, that was a pretty low moment.

posted by Blazecock Pileon at 2:16 PM on September 18, 2012

posted by Blazecock Pileon at 2:16 PM on September 18, 2012

1 is first, or the start, or the *prime* no matter the math, 1 begins.

posted by Mblue at 2:21 PM on September 18, 2012

posted by Mblue at 2:21 PM on September 18, 2012

> *When I was a kid, we were taught that 1 was a prime. *

Unless you are centuries old, you had poor math teachers using sloppy definitions.

posted by King Bee at 2:34 PM on September 18, 2012 [4 favorites]

Unless you are centuries old, you had poor math teachers using sloppy definitions.

posted by King Bee at 2:34 PM on September 18, 2012 [4 favorites]

> *1 is first, or the start, or the* prime *no matter the math, 1 begins.*

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 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.**

posted by King Bee at 2:38 PM on September 18, 2012 [8 favorites]

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 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.

posted by King Bee at 2:38 PM on September 18, 2012 [8 favorites]

This is like when they started adding more letters to the list of vowels.

posted by ceribus peribus at 2:44 PM on September 18, 2012

posted by ceribus peribus at 2:44 PM on September 18, 2012

Of course, the *largest* (known) prime is 2^43,112,609 - 1.

I think of it lurking waaay out there, malignant in its indivisibility, biding its time.

posted by Egg Shen at 2:46 PM on September 18, 2012 [2 favorites]

I think of it lurking waaay out there, malignant in its indivisibility, biding its time.

posted by Egg Shen at 2:46 PM on September 18, 2012 [2 favorites]

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.

posted by King Bee at 2:55 PM on September 18, 2012 [1 favorite]

posted by King Bee at 2:55 PM on September 18, 2012 [1 favorite]

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.

Not to mention plenty of other formulas, like the one Euclid himself devised for finding even perfect numbers.

posted by DynamiteToast at 2:55 PM on September 18, 2012

Pshh. Graham's Number doesn't even notice that Mersenne Prime as it casually grinds it into dust.

posted by kmz at 2:58 PM on September 18, 2012 [2 favorites]

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.

posted by michaelh at 2:59 PM on September 18, 2012

posted by michaelh at 2:59 PM on September 18, 2012

I guess by that point we'll be able to count using sub-sub-subatomic particles. There's hope!

posted by michaelh at 3:00 PM on September 18, 2012

posted by michaelh at 3:00 PM on September 18, 2012

kmz, one of my colleagues/friends and I often use such numbers to describe how many things we have to do before *next Tuesday* 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!

posted by King Bee at 3:04 PM on September 18, 2012 [8 favorites]

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!

posted by King Bee at 3:04 PM on September 18, 2012 [8 favorites]

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."

posted by baf at 3:36 PM on September 18, 2012 [3 favorites]

(Or, more simply and certainly: "The smallest prime number greater than TLPNTCBEWUUATAITU".)

posted by baf at 3:38 PM on September 18, 2012

posted by baf at 3:38 PM on September 18, 2012

(That won't necessarily be prime, but will have a larger prime divisor.)

posted by Obscure Reference at 3:39 PM on September 18, 2012 [3 favorites]

posted by Obscure Reference at 3:39 PM on September 18, 2012 [3 favorites]

posted by tigrefacile at 2:12 PM on September 18 [+] [!]

I'm pretty sure he was quoting a much earlier source.

posted by mykescipark at 3:44 PM on September 18, 2012

1 + (-1), 2 + (-2), 3 + (-3), 4 + (-4), ... ∞ + (-∞)

When Zero = Infinity (God's (!) Math)

posted by mrgrimm at 3:52 PM on September 18, 2012

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.

posted by JHarris at 3:57 PM on September 18, 2012 [7 favorites]

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.

posted by JHarris at 3:57 PM on September 18, 2012 [7 favorites]

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.

posted by King Bee at 4:03 PM on September 18, 2012

On preview, holy shit, JHarris. That last sentence you wrote is amazing.

posted by King Bee at 4:03 PM on September 18, 2012

Nomyte: *I understand why 0! = 1 is a convenient definition, but I'm amused by 0 choose 0 = 1.*

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.)

posted by stebulus at 4:03 PM on September 18, 2012

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.)

posted by stebulus at 4:03 PM on September 18, 2012

Put it back in your pants, gentlemen.

posted by Egg Shen at 4:16 PM on September 18, 2012 [3 favorites]

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.

posted by baf at 4:25 PM on September 18, 2012 [1 favorite]

posted by baf at 4:25 PM on September 18, 2012 [1 favorite]

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.

posted by King Bee at 4:28 PM on September 18, 2012 [1 favorite]

posted by King Bee at 4:28 PM on September 18, 2012 [1 favorite]

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!

posted by jamjam at 4:36 PM on September 18, 2012 [3 favorites]

posted by jamjam at 4:36 PM on September 18, 2012 [3 favorites]

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.

posted by stebulus at 4:46 PM on September 18, 2012

See Empty product 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)

posted by jepler at 4:47 PM on September 18, 2012

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)

posted by jepler at 4:47 PM on September 18, 2012

For me, the most interesting part of the original article is the bit where 1 is not a number at all.

posted by jepler at 4:49 PM on September 18, 2012 [1 favorite]

posted by jepler at 4:49 PM on September 18, 2012 [1 favorite]

The whole question is easily settled. Simply define 1 as a *dwarf* prime.

posted by Twang at 5:31 PM on September 18, 2012 [11 favorites]

posted by Twang at 5:31 PM on September 18, 2012 [11 favorites]

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.

*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.*

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. )

posted by leahwrenn at 8:06 PM on September 18, 2012

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. )

posted by leahwrenn at 8:06 PM on September 18, 2012

Randomly clicking around on wikipedia the other day I discovered smooth numbers, which actually have to do with their factorization. A number with no prime factors larger then some integerOne 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.

This of course leads to powersmooth numbers

posted by delmoi at 8:07 PM on September 18, 2012 [1 favorite]

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.

posted by stebulus at 8:44 PM on September 18, 2012

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.

posted by lubujackson at 10:00 PM on September 18, 2012

posted by lubujackson at 10:00 PM on September 18, 2012

Doing nothing is the easiest action of all!

posted by crazy_yeti at 11:21 PM on September 18, 2012

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.

Another way to get there:

"Erdos said it and I'll say it again,

There's always a prime between n and 2n."

posted by kaibutsu at 1:47 AM on September 19, 2012 [4 favorites]

Three Dog Night coined this in 1969.

posted by three blind mice at 2:16 AM on September 19, 2012

Words cannot express my feelings when I discovered that my daughter was being taught about the biological category 'mini-beasts' - worms, ants, centipedes, snails...

posted by Segundus at 2:33 AM on September 19, 2012 [1 favorite]

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:

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)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

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. [EWD641]

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". [EWD619]To see how this shakes out in mathematical practice, see, for example, Dijkstra's treatment of the infinitude of primes in EWD1203 and his criticism there of the proof as presented by Kac and Ulam, or his criticism in EWD993 of a statement on prime factorization. His criticisms would strike even many mathematicians as pedantic.

posted by stebulus at 12:35 PM on September 19, 2012 [4 favorites]

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posted by Philosopher Dirtbike at 1:47 PM on September 18, 2012 [4 favorites]