"There's actually no such thing as an uninteresting natural number"

October 16, 2013 6:58 AM

io9 takes a look at why the number 1729 shows up in so many Futurama episodes. It's mathtastic!
posted by quin (36 comments total)

At first I thought this post was a rather dull one, but then I noticed that

132890 = 3³+14³+23³+32³+44³ = 5³+11³+15³+35³+44³.
posted by Wolfdog at 7:08 AM on October 16, 2013 [39 favorites]

After he heard of the taxi-number incident, Ramanujan's collegue Littlewood said that "Every positive integer is one of Ramanujan's personal friends."
posted by ubiquity at 7:14 AM on October 16, 2013 [6 favorites]

I like reading about stuff like this even though I react to math textbooks the same way the apes in 2001 reacted to the monolith (minus the leap in evolution part).
posted by The Card Cheat at 7:15 AM on October 16, 2013 [1 favorite]

Linking the subject in case someone thinks it's just a cute quote. There is literally no such thing as an uninteresting natural number.
posted by Eideteker at 7:17 AM on October 16, 2013

The "interesting number paradox" is something of a joke, but it has a slightly less tongue-in-cheek analogue with an actual rigorous version, the Berry Paradox.
posted by roystgnr at 7:18 AM on October 16, 2013

quin's user number can also be written as a sum of five distinct, positive cubes, but it is not the smallest such number.
posted by Wolfdog at 7:22 AM on October 16, 2013

If you find this sort of thing interesting and want to read more, Waring's Problem [Wikipedia, MathWorld] is a good topic to look at. It leads to very difficult things very quickly. But that is true of asking questions about integers generally.

Increasingly, I believe it is a mistake to ever leave the field of two elements.
posted by Wolfdog at 7:28 AM on October 16, 2013 [2 favorites]

Lemma #0: The smallest natural number with property P is interesting.

Proof of Lemma #0: For a property of a natural number to be defined, there must be someone (S), somewhere who cared about that property at least once. (I seem to meet this person quite a lot. Frequently in a mirror.) Assume the smallest number with this property is not interesting. Point this out to person (S). Ask for a proof that you're wrong. There you go. Outsourcing is the future of mathematics, I tell you.....

Lemma #1: Being uninteresting is a property that could apply to a number.

Proof of Lemma #1: Existence proof: 34. I mean, really.

Proposition: "There's actually no such thing as an uninteresting natural number"
Proof:
1 is interesting. Maximum number of cats you can fit in a sack.
If a number N is the smallest uninteresting number, it must be 34.

Er, hang on. This seems to have gone wrong somewhere. No, don't leave, I'm just getting to the interesting bit....
posted by Combat Wombat at 7:32 AM on October 16, 2013 [4 favorites]

Applause for Wolfdog.
posted by Navelgazer at 7:37 AM on October 16, 2013 [1 favorite]

There is literally no such thing as an uninteresting natural number.

Here's a thought. Although we haven't defined "interesting", I think it's reasonable to say that if a number is interesting, then you should be able to write out the reason why it is interesting using, say, the English language and alphabet.

There are only countably many such explanations.

It follows that almost every real number is uninteresting.
posted by Wolfdog at 7:42 AM on October 16, 2013 [8 favorites]

"We're both expressible as the sum of two cubes!"
posted by crush-onastick at 7:45 AM on October 16, 2013 [4 favorites]

If numbers are everywhere and every number is interesting, then every integer is uninteresting because, really, after a while all that stimulation gets overwhelming and you just shut down.
posted by ardgedee at 7:46 AM on October 16, 2013 [1 favorite]

Case in point: 3487293595749930487
posted by sammyo at 7:52 AM on October 16, 2013

Remember: 2 is the oddest prime number because it is the only even prime number.
posted by eriko at 8:05 AM on October 16, 2013 [3 favorites]

Ook.
posted by mikelieman at 8:18 AM on October 16, 2013 [1 favorite]

There are only countably many such explanations.

It follows that almost every real number is uninteresting.

This only follows if reason_for_interesting(x) is one-to-one.
posted by a snickering nuthatch at 8:20 AM on October 16, 2013

It's got nothin' on 42, though.
posted by Thorzdad at 8:23 AM on October 16, 2013 [1 favorite]

I thought 1 was the smallest number expressible as the sum of two cubes: 0^3 + 1^3 = 1?
posted by Renoroc at 8:25 AM on October 16, 2013

You probably want to use "positive cubes", or throw the word "non-trivial" in there somewhere to make it interesting. Otherwise any cube is the sum of two cubes, itself and 0^3.
posted by Elementary Penguin at 8:27 AM on October 16, 2013 [1 favorite]

I thought 1 was the smallest number expressible as the sum of two cubes: 0^3 + 1^3 = 1?

Yes.

But 1729 is the smallest which can be represented as the sum of two cubes in two different ways.
posted by Wolfdog at 8:32 AM on October 16, 2013 [3 favorites]

Wha? I stumbled in here for the Futurama and now you want me to do math? Fie!
posted by Kitty Stardust at 8:36 AM on October 16, 2013 [2 favorites]

There is a story where a mathematician bets the devil that he cannot answer a question in 24 hours. The devil agrees. The mathematician say "Is [proof] true?" The devil says "what?" The mathematician says he wants to know if that particular proof is valid. The devil says "what does that even mean?" The mathematicians says "that is not my problem." The rest of the story is the devil popping in periodically throughout the day reporting on his progress. At midnight, he still doesn't have an answer, so the mathematician wins. The devil pays out, but hangs around. The mathematician asks him what he wants. The devil says "well, while I was trying to figure that out, I noticed this other interesting case, and I wonder if you had any ideas about it....

I suppose I could figure out what story that was, but I am afraid it would not be as good as I remember it....
posted by GenjiandProust at 8:36 AM on October 16, 2013 [1 favorite]

Simon Singh's latest book is about all the maths jokes buried in the Simpsons and Futurama, and this is one of the examples he's been talking about in interviews. This episode of the Pod Delusion ("a podcast of interesting things", with a Skpetical/Atheist slant) has a pretty good interview with him about it.
posted by metaBugs at 8:48 AM on October 16, 2013

Here's a thought. Although we haven't defined "interesting", I think it's reasonable to say that if a number is interesting, then you should be able to write out the reason why it is interesting using, say, the English language and alphabet.

There are only countably many such explanations.

It follows that almost every real number is uninteresting.

Objection: there are not countably many such explanations, there are infinite explanations.

-Suppose A is interesting for Reason 1.
-Suppose B is interesting for Reason 2.
-Let C=A+B
-Then C is interesting for Reason 3, which is that it is the smallest number that is the sum of two numbers that are interesting for Reason 1 and Reason 2.
-Let D=A+B+C
-Then D is interesting for Reason 4, which is that it is the smallest number that is the sum of three numbers that are interesting for Reason 1, Reason 2, and Reason 3.

ad infinitum
posted by googly at 9:12 AM on October 16, 2013

Well since there is no ordering of the real numbers, even the standard proof of interesting numbers fails on the reals (since it relies on the "smallest" uninteresting number). Uncountability is a helluva thing.
posted by Wulfhere at 9:14 AM on October 16, 2013

Well since there is no ordering of the real numbers, even the standard proof of interesting numbers fails on the reals (since it relies on the "smallest" uninteresting number). Uncountability is a helluva thing.

Nitpicking: The real numbers are ordered, just not well-ordered. If we assume the Axiom of Choice, however...
posted by bassooner at 9:23 AM on October 16, 2013

Objection: there are not countably many such explanations, there are infinite explanations.

You are correct that there are infinitely many explanations.

However, countable really is the appropriate word for certain kinds of infinite sets, including this one. This sentence from the linked article is the essence of it: "The elements of a countable set can be counted one at a time—although the counting may never finish, every element of the set will eventually be associated with a natural number."
posted by Wolfdog at 10:01 AM on October 16, 2013 [2 favorites]

It is the smallest number expressible as the sum of two cubes in two different ways.

I don't understand why anyone thinks this sort of thing is interesting. It's like the classic car aficionados who insist that their particular 1970 Camaro is *especially* rare because it's only "1 of 6 with factory air conditioning, the four-speed transmission, leather seats, cobalt blue paint and the chrome taillight trim" as if this completely arbitrary set of attributes somehow makes the car (or the integer) somehow more interesting.
posted by tylerkaraszewski at 10:09 AM on October 16, 2013

Bender and Flexo had taxi cab numbers (or both serial numbers were sums of two cubes) was also referenced in the series. Its that level of nerdy dedication that kept me watching the show.
posted by mrzarquon at 10:14 AM on October 16, 2013

It is the smallest number expressible as the sum of two cubes in two different ways....

I don't understand why anyone thinks this sort of thing is interesting.

If no one thought this sort of thing was interesting, much of the modern world [electronics, computers, cryptography] and humanity would not exist.
posted by Renoroc at 11:58 AM on October 16, 2013 [5 favorites]

0101100101? It's just gibberish.
posted by Slap*Happy at 12:49 PM on October 16, 2013 [2 favorites]

GenjiandProust - I believe that would be The Devil and Simon Flagg.
posted by tdismukes at 1:05 PM on October 16, 2013

At first I thought this post was a rather dull one, but then I noticed that

132890 = 91²+353² = 139²+337² = 157²+329² = 169²+323²
posted by MiltonRandKalman at 1:18 PM on October 16, 2013

My understanding is, the formal version of the "interesting number paradox" can be framed as a special case of Tarski's undefinability problem--it shows that it is not about the existence of interesting numbers, but rather that the predicate "interesting" has no consistent definition in any given (consistent) language.

My take-home from that is the hard truth that you can't have your cake and eat it and every number is a snowflake.
posted by polymodus at 1:37 PM on October 16, 2013

polymodus: another way to put it would be that "interesting" in terms of natural numbers would have to mean some variation of "possessing unique characteristics" and the whole thing with natural numbers is that each one is wholly unique from all others.
posted by Navelgazer at 4:56 PM on October 16, 2013

Jpfed: "There are only countably many such explanations.

It follows that almost every real number is uninteresting.

This only follows if reason_for_interesting(x) is one-to-one."

Yeah, though it's only wrong in the case that uncountably many real numbers are interesting for the same reason. As long as we're constructing fantasy functions, I think I like the assumption that this function is injective better.
posted by invitapriore at 9:04 PM on October 16, 2013