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I agree that this makes 37 somewhat interesting.

Certainly more interesting than - say - 31.

31 is a prime number too, and therefore somewhat interesting. But for sure not as interesting as 37, which as we just learned, is the median value for the second prime factor of an integer.

Any suggestions of integers which are even more interesting?

And while we are at it, is there an integer which qualifies to be the most interesting?

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Mathematicians are locked in bitter struggle between the Zeroastrians who believe additive identity blesses 0 as truly the most foundational and therefore interesting integer, while the Unitarians favor multiplicative identity and thus champion 1. Uncountable souls have been lost to the conflict between these two groups.

We Pragmaticists (some would say "abject pragmaticists") advocate for a compromise solution between Zeroastrians and Unitarians by taking h=½ as the fundamental constant. It satisfies several interesting identities involving other constants, like i^i=e^(-hπ), ∫e^(x^(1/h))dx=π^h, ∑n^(-1/h)=hπ^(1/h)/3, the celebrated inequality (a+b)h≥(ab)^h or the curious fact that 1/h is the smallest prime number.

Pragmaticists brought a fraction to an integer competition and lost

they lost the competition, but in the end it was them who paid the prize

Ah, but they won the Pi versus Tau competition ;-)

Halfasstrians?

Ugh, must we relate every thread about integer fascination back to Zerry/Unny zealotry? It’s been going on for centuries now, nobody’s changing their opinion at this point.

I, for one, think zero good will come of it…

Two soon?

> It’s been going on for centuries now, nobody’s changing their opinion at this point.

Sure, but one side can still out-breed the other.

because numbers have a natural meaning. roughly speaking each integer has as many meanings as its value. so 0 and 1 are just the very beginning.

zero's natural meaning is special, it's the void. sometimes the 'variable' stands in as having this meaning, sometimes the variable is zero.

one is the easiest simplest to grasp. its meaning is "I", ego.

two is duality in the broadest imaginable sense. wars are being fought right now over the precise meaning of 3

The Tao Produced One;

The One Produced Two;

The Two Produced Three

And The Three Produced All Things.

Sounds very close to the firstness, secondness and thirdness categories from Charles Sanders Peirce.

(Which I think are nothing but nonsense from a madman =)

He was an awfully pragmatic madman. He also discovered the universal and existential quantifiers independently of Frege, albeit a few years later.

No offense intended, but to the extent his work feels mad to you, it might be a “you problem.” That said, he never did reduce firstness, secondness, and thirdness to unity to his own satisfaction either. His work is truly tantalizing.

Oh and abduction has had some conceptual influence too.

Sad that the Arisbe fansite got deleted.

I agree this must be a "me" problem.

I just have never found these three "universal" categories useful for anything whatsoever except for a semiotics class in university where he was one of the study subjects.

Umberto Eco on the other hand... every single thing he wrote was immediately interesting and useful.

Or funny. His Lolita's parody, Granita, was Pratchett and Adams levels of funny. He was awesome.

we can be more precise now:

the three produced the equation, which pretended that all things were (could be) equalized into equivalence. they called this algebra and parised some monotheistic god; they called this magic and GOTO one.

thats funny i always heard it was an origin with two other orthogonal roles

unless youre a dirty filioquist

Perhaps 0 and 1 are the same. If there only exists the entity called 1, lacking all properties or relations to other entities, what is left? A void called 1, no different than 0. The true mystery is: how did plurality arise?

> Perhaps 0 and 1 are the same.

Compare https://en.wikipedia.org/wiki/Field_with_one_element

This feels closer to a tarot reading than mathematics, and I’m here for it

Consider for a moment that Aristotle, and thus most likely all the ancients, made a distinction between nothing and void.

My favorite is 2: Shared joy is a double joy, and shared sorrow is half sorrow.

I've always felt 2 was the most important as well, if only because for there to be something (1) there has to be nothing (0), and thus the existence of anything defines 2 states, being and non-being. Without the pair (0, 1) there is no distinguishing anything in the universe.

My spouse often claims that two is the best prime because she knows it annoys me. Or perhaps she is truly insane. Clearly 2 is the WORST prime, amirite?

Well, it's definitely the primest prime.

In modern math all numbers are built on set theory which has the empty set as it's foundation.

Empty set is more like 0 than 1. Check mate Unitarians.

The empty set is 1 set. Can’t have 0 without 1.

The empty set is the set with nothing, i e. no thing. So it doesn't need 1.

;)

>> The empty set is the set with nothing

So it is A set. In fact it's "the set" that makes it a rather special one.

Wouldn’t the set without nothing be more fundamental?

Brilliant. Go to the top of the class, er, set. Oops, forgot, sets have no order. List, then.

Ordered list, I mean.

Does the set of sets that exclude themselves exclude itself?

No, because it isn't a set, but rather a class.

A concept created at least partially to resolve this paradox.

That's a well-known old paradox, or close. Good on you, cobber. (AU slang :)

See Russells's paradox:

https://en.m.wikipedia.org/wiki/Russell%27s_paradox

You start from 1 empty set though.

If you could start from 0 empty sets, you'd have a point.

If I start from 0 empty sets, how do I have A point?

>You start from 1 empty set though.

Sez who, zigactly? If you are coldtea, I am coldertea. As in, I love my tea colder than you do yours. And starts who? Not me.

Don't kid around, dude. I dik around.

>If you could start from 0 empty sets, you'd have a point.

What if the set of empty sets has no element? :)

Was the first comment under the influence of drugs?

In any case, for the second comment: even if the set of empty sets has no elements, it's still ONE set.

>Was the first comment under the influence of drugs?

No, it was not. Regarding this whole subthread in which I participated, I did it for some mutual logic-chopping fun.

But if you thought there was a chance it was, you shouldn't have replied to the second comment, because I would have been equally under the influence of the drugs for the second, if I had been so for the first comment. (Don't reply to a druggie.) After all, they were both done in one atomic comment transaction.

No. Set theory is just one possible way you can build up numbers and mathematics.

You can just as well build up numbers and the rest of mathematics from different foundations. They are all equivalent.

Just like running your programs on ARM or on x86 or a toaster is all the same, thanks to emulators. (That's assuming you have an arbitrary amount of memory, and don't mind differences in runtime length.)

Which is more foundational: the wheel or the axle? Sets aren't worth much if you can't take unions, which is where 1 comes from.

0 and 1 are interesting as integers in the same way that a blank canvas is interesting as a painting. Very foundational, very useful but also very plain.

In my limited math experience, I really found an appreciation of the number one. My favorite part of high school algebra was realizing that "solving for x" was often just a repeated exercise of multiplying each side of the equation by a "clever form of one."

Solving for x often entails multiplying both sides by the same number, not necessarily one. Perhaps you're referring to simplifying fractions where you do often multiply by a clever form of one?

Ah, yes, you are absolutely correct. The _clever form of one_ trick did not apply as often as I recalled!

Pretty interesting if you work in an industry where you use them to represent every other number.

not in binary notation they're not

Must be an integer number of souls, though, so even in a conflict of infinite time, it won't be uncountable.

One of my professors would often refer to this struggle as taking place between the Nihilists and Unitarians.

Surely addition is more foundational than multiplication? Multiplication is just repeated addition.

Is this a reference for some fiction? Or you just made it up?

ℵ0 uncountable?

You missed the Dualitarians, the Trialitarians, and so on through to the Infinitarians ... whoops, forgot Cantor!

https://en.m.wikipedia.org/wiki/Georg_Cantor

> is there an integer which qualifies to be the most interesting?

I’d say that 664571016291591957042161991109590159107314773607 and 590488317782859198927092718316232684864014739572 qualify, which are the big- and little-endian versions of ASCII “the most interesting”, respectively.

https://en.wikipedia.org/wiki/Interesting_number_paradox

Like the Original Sin entered the Garden of Eden through the Serpent, mutable state breaks the time-invariant perfection of mathematics through the idea of interestingness, since it's clear the least interesting things only remain so until someone notices.

Well, that is about the question which is the most uninteresting number. And the paradox that - whichever number it is - this attribute makes it interesting.

But since we are not looking for the most uninteresting number, but the most interesting one, we do not have to fight with issues of that calibre.

True, but you must still contend with your observational influence. Whatever number you crown as "most interesting" will become even more interesting for wearing that crown!

So you must take this duty with extreme care and thoughtfulness, for when you anoint such a number, your action may crystalize it as such for all eternity. Other numbers will have to go further to supplant it.

I know it's a joke, but I still feel like it would be more meaningful if it wasn't so generic that it could kind of apply to anything. Like, there's nothing specific to numbers about the idea that the least-"interesting" item in some set is itself interesting for possessing that property.

Because it's numbers, which are sortable, you can ask which is the least member of a set. If a set of positive numbers doesn't have a least member, does the set exist?

There's a similar paradox which asks what the smallest number that can't be defined in fewer than thirteen words.

There must be some number, right? For instance, 7603201560, or "seven billion six hundred and three million two hundred and one thousand five hundred and sixty" seems to require 16 words. Maybe you can shorten it by removing the "ands," but I can come up with a longer number.

But if there were such a number, then couldn't we describe it as "the smallest number that can't be defined in fewer than thirteen words?" And then it just got defined in fewer than thirteen words. So it can't really be a member of that set.

So does the set have any members?

Then let's see who manages to write the least interesting comment in this thread.

The way I understood it, this formulation relies on induction on the natural numbers.

One other special property prime numbers can have is being irregular. And guess what the first irregular prime number is? 37. Interesting that it appears so late because irregular prime numbers do make up around 41% of all primes. See: https://encyclopediaofmath.org/wiki/Irregular_prime_number#:....

So the most interesting prime number really depends on what you consider more interesting. If you prefer the median second prime factor, then 37 is the best, but if you prefer the first irregular prime number then 37 is also the best. So it all depends on your perspective.

Another reason to like 37 is that it ends in a 7 so that it "sounds random" when someone asks for a number and it is better than 27 because it is prime. 7 is too low and 17 has connotations with bad luck. Although 37 is quite scary too: it's not just prime, which is pretty irregular, but it's also an irregular prime.

Historically, I'd say 60, because relative to it's size, it has many divisors (12), and among them a lot of useful ones (2, 3, 4, 5, 10), which is why our time system and trigonometry are probably based on it (360 = 6*60; 360 has 24 divisors).

I nominate 60. Babylonian mathematics was sexagesimal (base 60) which is a superior mathematical system to base 10. The number 60, a superior highly composite number, has twelve factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, of which 2, 3, and 5 are prime numbers. With so many factors, many common mathematical operations are much simpler than in base 10. Base 10 has no special mathematical properties apart from as an accident of evolution, we primates ended up with 10 fingers.

We still have some vestiges of the babylonian system however, 60 minutes in an hour. 360 (6 sixties) degrees in a circle.

It is another accident of evolution that we ended up with 12 finger joints, allowing one to count to twelve with a single hand by tapping the thumb on each finger segment in turn.

Since small prime factor are so useful, why not use 2 3 times and 3 2 times along with 5

  2*2*2 * 3*3 * 5 = 360

360 is a very handy number, it has a high degree of usefulness, when you're circumferential in your calculations.

You're perilously close to being a eikositriophile. We try not to talk about such things openly fnord.

https://en.wikipedia.org/wiki/23_enigma

How is that related?

I was being mostly frivolous but there's a larger point.

23 is used in also sorts of numerological explanations and connections. And of course, there is no actual connection 99% of the time.

The "explanations" get very inventive by also considering 2 and 3 invidually, 2+3, etc. Doesn't take long to get something to pull the gullible in.

Most of your argument seems to stand for 30 instead of 60: 2 x 3 x 5 = 30, which is a smaller number with mostly the same properties as 60.

If you want more divisibility (into quarters, eigths, ninths...), you can continue multiplying further, but 60 is as much a "historical accident" as 10 is an "accident of evolution".

60 is a 'highly composite' number (https://en.wikipedia.org/wiki/Highly_composite_number) - it is the lowest number with 12 divisors.

30 has divisors 1, 2, 3, 5, 6, 10, 15, 30 - eight divisors. But 24 is lower and also has eight (1, 2, 3, 4, 6, 8, 12, 24), making 30 less 'special'.

60 is the lowest highly composite number with 5 (and therefore 10) as a factor, which makes it a strong attractor when choosing bases for measuring systems.

Sure, but caring about 5 (and 10) is not what the GP comment stressed (actually, quite the opposite). If we do care about 10, then 60 is indeed appealing, but also an argument for base 10 too — which they argued against.

"Highly composite" number being used as a base for a numbering system does not really strike me as non-arbitrary either.

Choosing 60 and saying it makes more sense than, say, 2520 (which is also highly composite, but divisible by 1-10 too) is pretty arbitrary imho. Yes, my "criteria" is similarly arbitrary but focused on "10 does matter" ("I want to easily divide this into 2-10 groups"): both are too large to have individual names for each digit in a supposed number system with base 60 or 2520.

Much as I appreciate the reference, I think the fact this limit exists at all and is an integer is more interesting than what the integer is.

A lot of paradoxes and confused ideas begin with "choose an integer at random" - what is the average value of number chosen between 0 and infinity, for example.

> and is an integer

How could a median prime factor not be an integer?

The median value of [1, 2, 3, 4] is 2.5

But the median of [1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4] is 2. As the individuals observed values collect more and more repetitions—as they do when making a list of the second prime factor, for example—it becomes increasingly unlikely that the median will be between two values, not equal to one of the values.

It would be surprising if the median second prime factor was not a prime.

Sure, but 2 is the only even prime, so this is super constrained.

That's the mean.

And it’s also the median.

Depends if you are looking at a mean closed to integers or not: one can define a mean without rational or real numbers.

But it's not an integer before rounding. 37 is simply the nearest number to get approximately 50%.

I love that there could never be a least interesting natural number, because that fact would be interesting. In fact there can be no uninteresting natural numbers, because then you could list them, and the lowest one would be interesting for that fact.

17 is well known as the most random number between 1 and 20.

37 is also the smallest base (or second int argument) not supported by itoa(). Surely not a coincidence. ;)

It's because the characters in the result can be one of 10 digits or one of 26 letters.

Prime numbers are not especially interesting. E.g. all groups of prime size are cyclic. But each composite number has its own "character". The "interesting" number among primes is 2. Almost every theorem about prime numbers has to consider 2 as a special case.

Except that it's interesting that prime numbers have led inordinate numbers of individuals to devote uncountable but significants chunks of their lives investigating them.

It's not their fault that they're leftovers when an artificial grid is created by crossing-out every multiple of 2, 3, 5, or other prime numbers. Yet for some reason this seems to give them a fascinating aura for many. Maybe naming them 'prime' instead of 'leftover' is to blame.

I love the number 27 3^3 2+7 = 3+3+3

Very pretty. And to relate it somewhat to the topic: 1/27 = 0.037 repeating 037 and 1/37 = 0.027 repeating 027 :)

Every prime is odd and 2 is the oddest of them all!

The most interesting integer out there must certainly be the least interesting integer!

what is interesting to me is that we call imaginary integer numbers "Gaussian Integers", but we call the quaternion integer numbers "Hurwitz Quaternions" instead of "Hurwitz Integers".

Two is the oddest prime.

Smallest abundant number

12 is also the largest one-syllable number (at least in English).

I nominate 120 as "most interesting integer". For obvious reasons.

Um, it's not obvious to me. Could you specify?

The smallest triply perfect number? https://oeis.org/A005820

So many factors in such a small package, for one.

120 - or twelfty to friends - is a delight.

"The long hundred, also known as the great hundred or twelfty,[is the number 120 (in base-10 Hindu-Arabic numerals) that was referred to as hund, hund-teontig, hundrað, hundrath, or hundred in Germanic languages prior to the 15th century, and is now known as one hundred twenty, or six score."

https://en.wikipedia.org/wiki/Long_hundred

"120 is highly composite, superior highly composite, superabundant,[3] and colossally abundant. With sixteen divisors, 120 is the smallest number to have that many divisors..."

https://en.wikipedia.org/wiki/120_(number)

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