The most boring number in the world is...(scientificamerican.com)
scientificamerican.com
The most boring number in the world is...
https://www.scientificamerican.com/article/the-most-boring-number-in-the-world-is/
102 comments
> Makes me wonder if there's some forth dimension that humans are quantized to ignore
Interesting that your mind went to dimensionality - because the way to get 8 instead of "7" is to cut in 3D!
Do a + on the top, and then cut horizontally along one of the layers of icing! Voila! 8 cake pieces :D
Interesting that your mind went to dimensionality - because the way to get 8 instead of "7" is to cut in 3D!
Do a + on the top, and then cut horizontally along one of the layers of icing! Voila! 8 cake pieces :D
I attempted to do a cut in 4D to create 16 pieces, but I was only managed to have 8. The 8 pieces before I ate them and the 0 pieces after, for 8 total.
On a more serious (ish) note, can you cut a 3D pie in 4D space? If I go with a 2D circle in 3D space, by default you can't cut it down the extra dimension. But what if the circle is curved in 3D space, say by attempting to wrap it part the way around a sphere. Even if the circle was originally in the XY plane, you can now perform a Z plane cut which would allow going from 4 to 8 pieces. So ideally if you were to warp the 3D pie around a hypersphere, you would have a way of cutting it to make 16 pieces.
On a more serious (ish) note, can you cut a 3D pie in 4D space? If I go with a 2D circle in 3D space, by default you can't cut it down the extra dimension. But what if the circle is curved in 3D space, say by attempting to wrap it part the way around a sphere. Even if the circle was originally in the XY plane, you can now perform a Z plane cut which would allow going from 4 to 8 pieces. So ideally if you were to warp the 3D pie around a hypersphere, you would have a way of cutting it to make 16 pieces.
Based on what you're describing, I suppose you could have any number of pieces. Imagine you've got a 2D sheet of paper and you just fold it in half many times, and then cut through all the folds. You could have as many pieces as you want.
Good point. Stretching it against some wavy object and cutting would remove any meaning from the problem as you can get however many pieces you wanted based on the wave pattern.
This warping is equivalent to using a curved knife.
And if you think sufficiently hard about curved knifes and cherries on high-dimensional pies, you end up studying machine learning theory (https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_di...)
If you're implying that any problem of N-dimensional geometry can be reduced to machine learning techniques, then maybe there is a way to reinterpret machine learning as cutting N-dimensional cakes with M-dimensional knives?
Of course. There's a trivial proof that all neural networks can be reduced to a sufficiently complicated cake scenario. It's called Turing's Birthday Party.
Not _any_ problem, but the _specific_ problem of determining how many arbitrarily placed points (cherries) can be split by a given shape of hypothesis classes/classifiers (knives) is literally the definition of VC-dimension, yes :)
Nice!
Now I'm imagining a conspiracy among economists: to only ever show us top-down pie charts while funneling all the depth into their own slush fund. :)
Now I'm imagining a conspiracy among economists: to only ever show us top-down pie charts while funneling all the depth into their own slush fund. :)
[deleted]
The pattern of the 3 lines creating 7 pieces always reminded me of a poorly graffitied anarchy symbol. which, in its own fun way is ironic, as cutting pie that way is pure disorder and not forming the top vertex of the “A” along the circle does not fully conform to the standard anarchy symbol! (fwiw, I do know anarchy means no ruler/hierarchy rather than no rules/chaos, so pardon the relaxed definition in search of humor)
I felt similarly about the maximal number of regions you can obtain by joining N points on the circumference of a circle by straight lines. That sequence [1] starts:
1, 2, 4, 8, 16, 31
2^0
2^1
2^2
2^3
2^4
2^5 ... -1 ?I think it's more that stuff you deal with in normal life brings along substantial training in favor of fairness.
Dividing a shape means "divide fairly", because that's what you've done thousands of times. Drawing some lines means "in an aesthetically pleasing way" because that's what you've seen a billion times. Etc.
Math / what's possible doesn't care one bit about that though. It takes work to train yourself to move past other training.
Dividing a shape means "divide fairly", because that's what you've done thousands of times. Drawing some lines means "in an aesthetically pleasing way" because that's what you've seen a billion times. Etc.
Math / what's possible doesn't care one bit about that though. It takes work to train yourself to move past other training.
> I think it's more that stuff you deal with in normal life brings along substantial training in favor of fairness.
> Dividing a shape means "divide fairly", because that's what you've done thousands of times.
There are fundamental game-theoretic reasons for a fair split, yes. But there are also, arguably, more "pure" aspects at play, unrelated to agent dynamics. For example, out of the possible division ratios, there is an infinite number of them that give you two different pieces of cake, but only one ratio that gives you two indistinguishable pieces. What it means, among other things, is that[0] the 50/50 split is a singular one[1] that removes one bit of information from the "cake split into two pieces" system! Any other split requires you to track which piece is which (if only to be able to tell whether I just took the smaller or the larger one); the 50/50 one does not (the pieces are the same, it's pointless to ask which one I ate).
> Drawing some lines means "in an aesthetically pleasing way" because that's what you've seen a billion times. Etc.
The cake example above involves a special solution that removes bits from the problem. I feel the same is the case with a lot of aesthetics - we seem to have a preference for patterns that compress well. I don't think it's a peculiarity specific to human minds - it seems to me to be fundamental enough that all minds would naturally end up with a bias towards well-compressible pattern.
--
[0] - Take this piece of cake with a grain of salt, because I'm just a smartass, not an actual mathematician.
[1] - Ignoring the degenerate cases, i.e. the 0/100 and 100/0 splits.
> Dividing a shape means "divide fairly", because that's what you've done thousands of times.
There are fundamental game-theoretic reasons for a fair split, yes. But there are also, arguably, more "pure" aspects at play, unrelated to agent dynamics. For example, out of the possible division ratios, there is an infinite number of them that give you two different pieces of cake, but only one ratio that gives you two indistinguishable pieces. What it means, among other things, is that[0] the 50/50 split is a singular one[1] that removes one bit of information from the "cake split into two pieces" system! Any other split requires you to track which piece is which (if only to be able to tell whether I just took the smaller or the larger one); the 50/50 one does not (the pieces are the same, it's pointless to ask which one I ate).
> Drawing some lines means "in an aesthetically pleasing way" because that's what you've seen a billion times. Etc.
The cake example above involves a special solution that removes bits from the problem. I feel the same is the case with a lot of aesthetics - we seem to have a preference for patterns that compress well. I don't think it's a peculiarity specific to human minds - it seems to me to be fundamental enough that all minds would naturally end up with a bias towards well-compressible pattern.
--
[0] - Take this piece of cake with a grain of salt, because I'm just a smartass, not an actual mathematician.
[1] - Ignoring the degenerate cases, i.e. the 0/100 and 100/0 splits.
As long as no two lines are parallel and no three lines meet at a single point, you will get the maximum number of pieces.
You just have to be careful not to have two lines with similar slope, or you'll require a very large pie!
You just have to be careful not to have two lines with similar slope, or you'll require a very large pie!
Now I want to try this by placing the three cuts (chords) symmetrically so that they form a neat equilateral triangle at the center.
I wonder if it is possible to make just the right cut so all pieces are "equal" (in area)?
Is there a mathematical tool to make experiments like this graphically / visually? I define three cuts as lines that are offset by "x" from the center of the circle and as I vary the value of x from zero to r, the areas of each cut piece should be computed and plotted.
I wonder if it is possible to make just the right cut so all pieces are "equal" (in area)?
Is there a mathematical tool to make experiments like this graphically / visually? I define three cuts as lines that are offset by "x" from the center of the circle and as I vary the value of x from zero to r, the areas of each cut piece should be computed and plotted.
Better not look at the related sequence https://oeis.org/A000127
1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386...
1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386...
> Maximal number of regions obtained by joining n points around a circle by straight lines.
I was ready to attempt drawing that, but then I saw the other possible definition:
> Also number of regions in 4-space formed by n-1 hyperplanes.
So much easier to visualize it like this!
I was ready to attempt drawing that, but then I saw the other possible definition:
> Also number of regions in 4-space formed by n-1 hyperplanes.
So much easier to visualize it like this!
> I even drew it, and it was still damned difficult to make my hand offset the line to leave that shit piece in the middle!
Make the offset bigger and then the middle piece can be plenty big!
Make the offset bigger and then the middle piece can be plenty big!
It felt more natural after I renamed it the "inept" or "a*hole" caterer's sequence.
I wonder how much of this is due to a cultural sense of fairness and what proper pie cutting looks like.
A friend mentioned being at a party once where the guest of honor cut the first slice for themself and they chose to make a glancing slice that maximized their surface frosting to volume ratio.
I wonder how much of this is due to a cultural sense of fairness and what proper pie cutting looks like.
A friend mentioned being at a party once where the guest of honor cut the first slice for themself and they chose to make a glancing slice that maximized their surface frosting to volume ratio.
> > or the “lazy caterer’s sequence” (1, 2, 4, 7, 11, 16, 22, 29,...), the maximum number of pie pieces that can be achieved by n cuts.
> My brain just did not want to accept "7" as the fourth number in the series.
The sequence that you get from the difference of successive numbers in the list is rather predictable: 1,2,3,4,5,6,7, ...
> My brain just did not want to accept "7" as the fourth number in the series.
The sequence that you get from the difference of successive numbers in the list is rather predictable: 1,2,3,4,5,6,7, ...
A picture of a pancake cut into 7 pieces in 3 cuts:
https://en.m.wikipedia.org/wiki/File:PancakeCutThrice.agr.jp...
https://en.m.wikipedia.org/wiki/File:PancakeCutThrice.agr.jp...
<Looks at Wikipedia>
Lazy Caterer make Grug angry. Grug smash stupid caterer in stupid face.
Lazy Caterer make Grug angry. Grug smash stupid caterer in stupid face.
i think this is a simple consequence of euler characteristic, and the property that any addition cut intersects each of the others at most once. You can set up a sinple equation that gives you an upper bound and a simple configuration which achieves it.
This article seems aimed to resolve the "Interesting Number Paradox"[0], but does so in a pretty weak way. The author concludes:
> So the split into interesting and boring numbers seems to stem from the judgments we make, such as attaching importance to prime numbers.
Given the context of the rest of the article, about what numbers appear in OEIS[1] sequences, I'd argue that the author has assumed the conclusion from the start. If you look only at numbers that people have deemed interesting, you....only find numbers that people have deemed interesting.
There's a much deeper (maybe-philosophical) question left unasked about the nature and structure of numbers. I was hoping the author would at least wave to that.
[0] https://en.wikipedia.org/wiki/Interesting_number_paradox
[1] https://oeis.org/
> So the split into interesting and boring numbers seems to stem from the judgments we make, such as attaching importance to prime numbers.
Given the context of the rest of the article, about what numbers appear in OEIS[1] sequences, I'd argue that the author has assumed the conclusion from the start. If you look only at numbers that people have deemed interesting, you....only find numbers that people have deemed interesting.
There's a much deeper (maybe-philosophical) question left unasked about the nature and structure of numbers. I was hoping the author would at least wave to that.
[0] https://en.wikipedia.org/wiki/Interesting_number_paradox
[1] https://oeis.org/
I think there's one more deep, interesting (!) question hidden, adjacent to the post and your reply: what exactly makes something interesting? I feel it's a real question, because the things we find interesting aren't all that arbitrary.
I have only a handwavy notion of it, based on some on-line articles (IIRC one of them might have been a blogpost or a paper by Scott Aaronson), but the general idea is that the things we find interesting exist in the middle between low and high-entropy states.
For example, take matches. A box of matches is relatively low-entropy: they all sit there neatly arranged in parallel rows, pointing the same direction. Matches in a box are boring. Now, dump the box onto the table. The matches are now arranged in a random pattern. This is a high-entropy state, and it's even more boring than the low-entropy one. Now, start putting those matches into some pattern - suddenly, things get interesting. One of the particularly interesting configurations would be a self-supporting 3D structure made of those matches - and that's, in terms of entropy, about half-way between matches in a box, and matches on a pile.
Put in information-theoretic terms, it seems that we find both highly predictable and highly unpredictable configurations boring: the former likely because there is just not much new information to pay attention to - but the latter, seemingly because it's too unpredictable, and we round it up to "pure noise".
So my question is, is there any formal description of this concept? A mathematical answer to why we find both order and disorder boring, and are most interested in things in-between? Some kind of natural quantity that tracks -E² (with E being entropy), and which would be fundamentally useful to maximize?
I have only a handwavy notion of it, based on some on-line articles (IIRC one of them might have been a blogpost or a paper by Scott Aaronson), but the general idea is that the things we find interesting exist in the middle between low and high-entropy states.
For example, take matches. A box of matches is relatively low-entropy: they all sit there neatly arranged in parallel rows, pointing the same direction. Matches in a box are boring. Now, dump the box onto the table. The matches are now arranged in a random pattern. This is a high-entropy state, and it's even more boring than the low-entropy one. Now, start putting those matches into some pattern - suddenly, things get interesting. One of the particularly interesting configurations would be a self-supporting 3D structure made of those matches - and that's, in terms of entropy, about half-way between matches in a box, and matches on a pile.
Put in information-theoretic terms, it seems that we find both highly predictable and highly unpredictable configurations boring: the former likely because there is just not much new information to pay attention to - but the latter, seemingly because it's too unpredictable, and we round it up to "pure noise".
So my question is, is there any formal description of this concept? A mathematical answer to why we find both order and disorder boring, and are most interested in things in-between? Some kind of natural quantity that tracks -E² (with E being entropy), and which would be fundamentally useful to maximize?
Thank you for your great comment, I made an account just to reply. Please see Robert Sapolsky, Stanford Class Day 2009 lecture on "Uniqueness of Humans". He shares experiments on primates involving dopamine, which closely mirror your comment on entropy buy without using the term.
If you can find it, please share the blog or paper where you found the idea.
If you can find it, please share the blog or paper where you found the idea.
I figure both order and disorder compress easily, whereas stuff in the middle doesn't. A box of matches describes the situation pretty succinctly. A pile of matches is also pretty succinct. A number of matches arranged in the shape of a dragon fractal is getting harder to denote succinctly.
I think this more or less has to do with formal languages for specifying numbers. If you have a formal language in which any string might (or might not) be interpreted to mean a single integer, then by the pigeonhole principle, half the integers of a given string-length will require strings of greater information length expression to be referred to.
So relative to this given formal language, you can have a smaller integer that doesn't have a compact representation, doesn't have representation "smaller" than itself (and so might be labeled "boring"). And you can refer to the smallest of these integers in some meta-language but you can't refer to this smallest integer within the given formal language (and still have the language be consistent).
Related to: Kolmogorov complexity, Godel's Second Incompleteness Theorem, Chaitin's Number Omega
So relative to this given formal language, you can have a smaller integer that doesn't have a compact representation, doesn't have representation "smaller" than itself (and so might be labeled "boring"). And you can refer to the smallest of these integers in some meta-language but you can't refer to this smallest integer within the given formal language (and still have the language be consistent).
Related to: Kolmogorov complexity, Godel's Second Incompleteness Theorem, Chaitin's Number Omega
Super weird to me that the OEIS only stores "first 180 or so characters of a number sequence". Of course that's going to completely mess up the analysis in the article. They do mention this. There's something strange to me about how often analysis doesn't have disclaimers like "0.1% of data wasn't available but we think that's fine" but more like "the writing samples were collected on napkins with Sharpie in a Denny's at 4am and every sample from someone with a last name starting the letters M-Z was lost in a fire" then goes on to make a bunch of conclusions about modern essay writing or something.
Anyway, it'd be cool if the OEIS had the numbers up to some limit. Maybe 2^16 so it fits into two bytes. It would be at most 1.2MB per entry, but generally way way smaller.
Anyway, it'd be cool if the OEIS had the numbers up to some limit. Maybe 2^16 so it fits into two bytes. It would be at most 1.2MB per entry, but generally way way smaller.
It’s not actually true; the OEIS is capable of storing a lot more numbers per entry in the form of a “B-file”, for example: https://oeis.org/A000127/b000127.txt. You can see the documentation on this feature here: https://oeis.org/wiki/B-files.
As far as I understand it, though, these B-files aren’t indexed by the search engine.
As far as I understand it, though, these B-files aren’t indexed by the search engine.
Thanks! I looked around and didn't see that.
Do you know that not every sequence is easily computable ?
Like, most of them isn't.
Even if you have an algorithm to compute them. I appreciate your optimism thought. But OEIS is made by pretty smart people, if it was so simple, it would have been made.
It would be great to make it when possible but the bias will remains.
Even if you have an algorithm to compute them. I appreciate your optimism thought. But OEIS is made by pretty smart people, if it was so simple, it would have been made.
It would be great to make it when possible but the bias will remains.
> if it was so simple, it would have been made
I don't believe this to be true. Maybe the people who make OEIS have been satisfied with having the description and a few examples.
It would only work well for strictly increasing sequences. I don't know of any sequence that increases slowly and is difficult to compute. If it increases quickly, then you'll quickly pass the limit. If it's not increasing, then you'd have to prove which numbers are under the limit and that could be very challenging in some cases.
I don't believe this to be true. Maybe the people who make OEIS have been satisfied with having the description and a few examples.
It would only work well for strictly increasing sequences. I don't know of any sequence that increases slowly and is difficult to compute. If it increases quickly, then you'll quickly pass the limit. If it's not increasing, then you'd have to prove which numbers are under the limit and that could be very challenging in some cases.
The first thing that came to mind for slowly increasing worth looking into seemed like it might be the inverse ackermann function, which curiously I didn't find any OEIS entry for.
Inverse ackerman should be easy to compute by computing ackerman for enough terms and inversing numerically?
I mean, it will be the same number basically for a looooong time
I don't think they're criticizing the OEIS's approach for storing sequences, but the person who decided to use their prefixes to draw conclusions about which numbers are "boring".
> Super weird to me that the OEIS only stores "first 180 or so characters of a number sequence"
I agree! Another opportunity for a paper though!
If f(180) = y = 20067, what does the graph of f(x) look like?
How quickly would f(x) grow? Linearly? Quadratic? Exponential? Logarithmic? Any bets?
I agree! Another opportunity for a paper though!
If f(180) = y = 20067, what does the graph of f(x) look like?
How quickly would f(x) grow? Linearly? Quadratic? Exponential? Logarithmic? Any bets?
The problem is that f needs to be interesting. Nobody really cares about the function "17393/16200 x^2 - 14719/180 x", even if that produces f(180) = 20067.
You misunderstood my point... f in this example is the function which computes the first number that does not show up in the first X samples of OEIS sequences(so in the paper cited, OEIS stores only the first 180, so when x is 180, f(x) = 20067).
Interestingly, this will create a paradox because it could itself be expressed as an OEIS sequence, which would mean that it does not belong in the sequence... but if it doesn't belong, then it should belong(Russell's paradox anyone :D).
Interestingly, this will create a paradox because it could itself be expressed as an OEIS sequence, which would mean that it does not belong in the sequence... but if it doesn't belong, then it should belong(Russell's paradox anyone :D).
As far as I'm aware, most OEIS sequences have a variety of different algorithms (Maple, Mathematica) to generate them. It would be nice to be able to extend these sequences out and see if this has any change to "Sloane's gap".
Try this one: https://oeis.org/A060843
It's the Busy Beaver problem and BB-5 is proven ≥ 47176870. The list here would be the full list for any "up to n" database smaller than that.
I'm just highlighting that some sequences are not so easily generated. BB(n) is canonically awkward in this regard.
> [1729] is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.
This isn’t actually correct. 1729 is the smallest number expressible as a sum of two positive cubes in two different ways. The smallest number expressible as a sum of two cubes in two different ways is 91 (3^3+4^3 and -5^3+6^3).
This isn’t actually correct. 1729 is the smallest number expressible as a sum of two positive cubes in two different ways. The smallest number expressible as a sum of two cubes in two different ways is 91 (3^3+4^3 and -5^3+6^3).
> The smallest number expressible as a sum of two cubes in two different ways is 91 (3^3+4^3 and -5^3+6^3).
That can't be true either since -91 = (-3)^3 + (-4)^3 and -91 = 5^3 + (-6)^3. Since -91 < 91, the statement has to be false.
In fact, it is not the number x with smallest value |x| that can be expressed as the sum of two cubes in two different ways since, 0 = (-1)^3 + 1^3 and 0 = (-2)^3 + 2^3.
It appears that 91 is the smallest positive integer that can be expressed as a sum of two cubes in two different ways.
There is no value to this comment. I mostly just wanted to be annoying. I hope I'm right, at least.
That can't be true either since -91 = (-3)^3 + (-4)^3 and -91 = 5^3 + (-6)^3. Since -91 < 91, the statement has to be false.
In fact, it is not the number x with smallest value |x| that can be expressed as the sum of two cubes in two different ways since, 0 = (-1)^3 + 1^3 and 0 = (-2)^3 + 2^3.
It appears that 91 is the smallest positive integer that can be expressed as a sum of two cubes in two different ways.
There is no value to this comment. I mostly just wanted to be annoying. I hope I'm right, at least.
> There is no value to this comment. I mostly just wanted to be annoying
Since annoying provides negative value, your statement is incorrect.
Since annoying provides negative value, your statement is incorrect.
Oh no! Hoist by my own petard of pedanticity! XD
> There is no value to this comment.
There is always value in truthful statements. You are right that 91 is not the smallest number as was implied by the parent. -91 is. After all it was the parent who brought in the negative integers into the mix.
There is always value in truthful statements. You are right that 91 is not the smallest number as was implied by the parent. -91 is. After all it was the parent who brought in the negative integers into the mix.
> There is always value in truthful statements.
As long as we're not talking about nonzero value.
I think GP made a worthwhile point, but in general the universe is full of uninteresting truths. Like take two large, distinct numbers, assert that they are unequal. Is that specific truth valuable?
As long as we're not talking about nonzero value.
I think GP made a worthwhile point, but in general the universe is full of uninteresting truths. Like take two large, distinct numbers, assert that they are unequal. Is that specific truth valuable?
Is that still true with complex or higher dimension numbers in general?
+832375400086541 if you can provide a proof through mathematical induction.
+832375400086541 if you can provide a proof through mathematical induction.
There's no total order on complex numbers, so what would smallest mean there? The numbers z=x+iy (x,y in Z) with smallest positive |z|? Fair enough.
You make an important point, but I understood the parent as, "what is the smallest integer expressible as the sum of two cubes of Cayley-Dickson algebras of order n?"
So you're allowed to use complex/quaternions etc. in the cubes, but the number has to come from those well-ordered integers.
So you're allowed to use complex/quaternions etc. in the cubes, but the number has to come from those well-ordered integers.
>theres no total order on the complex numbers
Sure there is, lex order on Re and Im and then inherit from the reals.
Sure there is, lex order on Re and Im and then inherit from the reals.
Aren’t the numbers that appear most often in the OIES also boring? I think a number that doesn’t appear in any sequences automatically promotes it to interesting; it’s some kind of sequence prime without being an actual prime.
Also, there is a catalog of interesting numbers: the Penguin Dictionary of Curious and Interesting numbers. A college prof of mine would read an entry before each class. https://www.thriftbooks.com/w/the-penguin-dictionary-of-curi...
Also, there is a catalog of interesting numbers: the Penguin Dictionary of Curious and Interesting numbers. A college prof of mine would read an entry before each class. https://www.thriftbooks.com/w/the-penguin-dictionary-of-curi...
Reminds me of this Numberphile video, which might be interesting to anyone wanting to keep following this rabbit hole.
https://youtu.be/VDYzSzDaHuM
https://youtu.be/VDYzSzDaHuM
This article reminds me a lot of this false proof about describing numbers with words [0] (although I first encountered it relating to integers, not real numbers).
[0] https://jeremykun.com/2011/07/28/false-proof-twenty-word/
[0] https://jeremykun.com/2011/07/28/false-proof-twenty-word/
In 7th grade I went to a program for kids with advanced test scores, and met a kid who talked about math the entire time. He challenged me by asking what the most boring number was, and I was very pleased with myself to say that the most boring number couldn’t exist because it would therefore be interesting due to that trait.
The interaction and his name stuck in my mind, even as I entered an era of underachievement. Today he has a handful of degrees, including a PhD/DPhil in Mathematics at Oxford. He has no recollection of or interest in our short meeting that remains lodged in my memory.
The interaction and his name stuck in my mind, even as I entered an era of underachievement. Today he has a handful of degrees, including a PhD/DPhil in Mathematics at Oxford. He has no recollection of or interest in our short meeting that remains lodged in my memory.
Who else clicked the link thinking the answer would be easy to find only to get sucked down a rabbit hole?
Me.
> 20,067
Because it’s the smallest number not in any of the sequences they describe. It wasn’t worth the ride.
> 20,067
Because it’s the smallest number not in any of the sequences they describe. It wasn’t worth the ride.
The scene is depicted in the film "The Man Who Knew Infinity", highly recommended.
https://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinity
https://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinity
The larger numbers get, the more boring they are. 385734874675783289473 is really boring.
Not true, that number is immortalized in this famous HN thread: https://news.ycombinator.com/item?id=35059068
Agreed. https://news.ycombinator.com/item?id=35059068.38573487467578... is also great for obvious reasons.
It's interesting that you can tell this is a number row smash. Here's a histogram of frequency.
X
X XX
XX XX
XXX XX
XXXXXXXX
1234567890Being able to tell that is interesting! It does not make the number interesting. There's so many equivalent smashes.
If we're going to use interesting in the sense in the article, that just means it has some attribute. We could formalize keyboard smash and this number would have that. If it's interesting in the common sense, then it's subjective. I found this number interesting enough to spend a few minutes on it, which is more than I give most numbers!
> some attribute
More than that. Every number is either even or odd! Super common attributes don't do enough.
> I found this number interesting enough to spend a few minutes on it, which is more than I give most numbers!
I would say you didn't find the number interesting, you found the context interesting. There are billions of similar numbers that could have gone in the same slot and you would have spent the same effort on them. But if you were set free to look for interesting numbers, without being fed this one, you never would have glanced at it. And more specifically, this is a context that would basically never produce the same number again; it's not a sequence with a generator.
More than that. Every number is either even or odd! Super common attributes don't do enough.
> I found this number interesting enough to spend a few minutes on it, which is more than I give most numbers!
I would say you didn't find the number interesting, you found the context interesting. There are billions of similar numbers that could have gone in the same slot and you would have spent the same effort on them. But if you were set free to look for interesting numbers, without being fed this one, you never would have glanced at it. And more specifically, this is a context that would basically never produce the same number again; it's not a sequence with a generator.
The more we keep talking about it, the more interesting I find this number. It seems to have quite the ability to generate discussion. It may be the least boring number!
Let's imagine we have a ton of people try to come up with interesting numbers.
Half of them try this keysmash method, and half of them try to come up with numbers without such randomness.
Then we have them each vote for a number they find interesting.
In that case, numbers like the above will get one vote. Lots of other numbers in these sequences will have tens, hundreds, thousands of votes.
Are the votes wrong? Is your number really the most interesting? Is it even the most interesting number between n-10 and n+10?
Throwing out half the resumes you get won't help you pick a candidate with better luck. Happening to randomly generate a number once doesn't make it interesting.
Nothing in this discussion is specific to that number. All the interestingness of this discussion should be split trillions of ways across all the similar keysmash numbers.
Half of them try this keysmash method, and half of them try to come up with numbers without such randomness.
Then we have them each vote for a number they find interesting.
In that case, numbers like the above will get one vote. Lots of other numbers in these sequences will have tens, hundreds, thousands of votes.
Are the votes wrong? Is your number really the most interesting? Is it even the most interesting number between n-10 and n+10?
Throwing out half the resumes you get won't help you pick a candidate with better luck. Happening to randomly generate a number once doesn't make it interesting.
Nothing in this discussion is specific to that number. All the interestingness of this discussion should be split trillions of ways across all the similar keysmash numbers.
Calling it the least boring number was a joke, really. I do maintain two points seriously. First, that the digit frequency in this number reveals how it was generated and that characteristic lifts it off the floor of "least interesting numbers". Yes, there are a lot of numbers in this keyboard smash group. There are also a lot of primes. Second, I am going to give some credit to this number, 385734874675783289473. You can come up with a hypothetical, describe what might have happened in other universes. But, that's not what actually happened. Not all of the interestingness should be split because only one of those numbers actually showed up in this discussion in this universe. There's something unromantic about forgetting that, like a man sending a small anniversary present to every woman he went to college with because maybe he'd have ended up marrying any of them.
Well, the people you meet only exist in this one lifetime. But the numbers themselves transcend that. I think it's okay to attach a tiny bit of significance to a number that happens to be randomly generated, but I don't think it's enough to move a number up to "interesting" unless it was right on the edge.
Brilliant! You are correct sir. Sometimes HN delivers ;-)
Its sum of divisors is 385735449676045680000, very close to the original number. In base ten that shares the first five digits with the number itself. It's also small since the expected value is the number itself times π²/6, whereas our sum of divisors is slightly less than the original number. This is probably due to its only having three prime factors, all multiplicity 1. That's unusual too but estimating what we would expect is awkward.
edit: I should add that all of this was well known to Ramanujan, with whom I share a birthday. He famously was able to refute the uninterestingness of 1729.
edit: I should add that all of this was well known to Ramanujan, with whom I share a birthday. He famously was able to refute the uninterestingness of 1729.
How did you figure out all of that? Surely, you don't mean that Ramanujan knew about 385734874675783289473 because the sibling comment shows that it was keyboard-smashed.
Whatever the number theory equivalent of keyboard mashing is. I just checked a few typical functions and that one seemed interesting. Ramanujan wrote his thesis (equivalent) on highly composite numbers and studied things like the "divisor functions" related to summing (powers of) a number's divisors. I'd almost be ready to say he wouldn't have had anything to say about Lacker's Number, but then he was responsible for quite a few results which he had no reasonable way of knowing, given the knowledge available.
Someone should submit a series of "boring numbers" to the OEIS. It would include all numbers not listed in any other series. Although it would be hard keeping it up-to-date.
THEOREM: All numbers are interesting.
PROOF BY CONTRADICTION: If there are boring numbers, then there must be a minimum absolute boring one; which makes it an interesting number. QED
PROOF BY CONTRADICTION: If there are boring numbers, then there must be a minimum absolute boring one; which makes it an interesting number. QED
If you take an infinite pile of whole, positive integer numbers and sieve out all of the interesting numbers that are interesting for reasons other than they are not interesting in any other way, you would be left with a smaller infinite pile of numbers that are only interesting because they are not interesting in any way other than that they are in the pile of non-interesting numbers.
Therefore, there is an infinite amount of numbers that are not interesting for any other reason than that they are not interesting.
The fact that the most obviously not interesting number in the infinite pile of uninteresting numbers is interesting because it is so obviously uninteresting does not distract from the infinite pile of not interesting numbers below it.
We can assign that uninteresting number a value of 1/1.
If the second most obviously least interesting number is interesting because it is the second least interesting number, then we can say that it has an interest value of 1/2 of the most obviously least interesting number.
The next one would get 1/3, and the next 1/4, and so on and so on through 1/infinity, which converges to mean that there is a total of 2 points of interest in the uninteresting group. (The first is that there is a most least interesting number, and the second in that there is an infinite pile of uninteresting numbers aside from that).
I'm gonna do some hand waving here and say that this means that there is a point where the number are so large and incomprehensible to human or machine minds, where the values of the numbers are so entirely lost in computational morass that to assign the nomenclature that differentiates that number from its brethren requires more computational power than there will ever be atoms in the universe capable of being used for computation between the beginning and end of time.
Somewhere out past that is a number whose 1/"interest" value is so denominatorily close to infinity that it is, in fact, the least interesting number even when taking into account that the least interesting number is interesting for its uninterestingness. Remember, we are already working within the infinite set of fundamentally uninteresting numbers.
Therefore, there is at least one uninteresting number that is completely boring and uninteresting despite the fact that uninteresting numbers are interesting because of their uninterestingness.
That number must be the most uninteresting number regardless of whether it is interesting because of its lack of interestingness because there has already been an infinite number of interestingly uninteresting numbers to the point where there is no interest left to account for its uninterestingness.
This number would be so uninteresting as to be impossible to even think of. The moment you can conceptualize a number it would automatically be excluded from the list of candidates for the most perfectly uninteresting number. Even the concept that this response has generated has created a false concept of the least interesting number. The least interesting number is beyond this explanation and the abstract concept of a "number" that this conveys, the number that you could say would be interesting because of its uninterestingness, is not the number that is being expressed here. It is too uninteresting to be identified as anything concrete, too elusive to be conceptualized, too mystic to be grasps, too meta to be connected to the tangible universe, and there are so many of them you could hold every infinite uninteresting number that may or may not possibly exist in the aplm of your hand and you still wouldn't be able to grasp it because of how uninteresting it is, even though the description of how difficult the number is to express how uninteresting the number is is interesting, the number itself means nothing either directly or by comparison.
Therefore, there is an infinite amount of numbers that are not interesting for any other reason than that they are not interesting.
The fact that the most obviously not interesting number in the infinite pile of uninteresting numbers is interesting because it is so obviously uninteresting does not distract from the infinite pile of not interesting numbers below it.
We can assign that uninteresting number a value of 1/1.
If the second most obviously least interesting number is interesting because it is the second least interesting number, then we can say that it has an interest value of 1/2 of the most obviously least interesting number.
The next one would get 1/3, and the next 1/4, and so on and so on through 1/infinity, which converges to mean that there is a total of 2 points of interest in the uninteresting group. (The first is that there is a most least interesting number, and the second in that there is an infinite pile of uninteresting numbers aside from that).
I'm gonna do some hand waving here and say that this means that there is a point where the number are so large and incomprehensible to human or machine minds, where the values of the numbers are so entirely lost in computational morass that to assign the nomenclature that differentiates that number from its brethren requires more computational power than there will ever be atoms in the universe capable of being used for computation between the beginning and end of time.
Somewhere out past that is a number whose 1/"interest" value is so denominatorily close to infinity that it is, in fact, the least interesting number even when taking into account that the least interesting number is interesting for its uninterestingness. Remember, we are already working within the infinite set of fundamentally uninteresting numbers.
Therefore, there is at least one uninteresting number that is completely boring and uninteresting despite the fact that uninteresting numbers are interesting because of their uninterestingness.
That number must be the most uninteresting number regardless of whether it is interesting because of its lack of interestingness because there has already been an infinite number of interestingly uninteresting numbers to the point where there is no interest left to account for its uninterestingness.
This number would be so uninteresting as to be impossible to even think of. The moment you can conceptualize a number it would automatically be excluded from the list of candidates for the most perfectly uninteresting number. Even the concept that this response has generated has created a false concept of the least interesting number. The least interesting number is beyond this explanation and the abstract concept of a "number" that this conveys, the number that you could say would be interesting because of its uninterestingness, is not the number that is being expressed here. It is too uninteresting to be identified as anything concrete, too elusive to be conceptualized, too mystic to be grasps, too meta to be connected to the tangible universe, and there are so many of them you could hold every infinite uninteresting number that may or may not possibly exist in the aplm of your hand and you still wouldn't be able to grasp it because of how uninteresting it is, even though the description of how difficult the number is to express how uninteresting the number is is interesting, the number itself means nothing either directly or by comparison.
Wouldn't the smallest boring number be interesting, and therefore not a boring number? And then the logical conclusion would be that there are no boring numbers.
If there are boring number, then your assertion holds. But if there are no boring numbers, then it doesn't and therefore there are boring numbers.
How long until there’s an automated daily submission to OEIS of the first 180 integers not covered by an OEIS sequence.
It would be rejected, the OEIS is for sequences with a well-defined mathematical meaning.
{ a∈ℤ | ∀S∈O a∉S }
You'd have to clarify to "OIES as of March 8th, 2023, at midnight GMT" (or something similar).
Then you need to import a full copy of that OIES (as definitions should be self-contained as much as possible). That importing is what would stop you -- the definition of the problem would end up far too huge.
Then you need to import a full copy of that OIES (as definitions should be self-contained as much as possible). That importing is what would stop you -- the definition of the problem would end up far too huge.
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I wonder if the gap has narrowed over time.
TL;DR It’s 20,067
806179
BORIng
And them, there is 241543903.
TLDR. So what is the actual number?
Prime numbers are interesting in this regard not for the notional reason (low Kolmogorov complexity) but because if their high “randomness.” They are not patterned so much as they are the leftovers excluded by patterns (notably, nontrivial multiplications) but, for this reason, of interest to number theorists simply because facile pattern “shouldn’t be” there. Which I guess gets to the heart of why diagonal arguments so easily torpedo discussions of “interestingness” or the lack thereof.
One of my favorite stories concerning primes is how fermat thought he discovered the formula for primes ( Fermat's primes ). He only checked his formula for the first few primes and assumed it worked for all primes. Euler checked for the next prime using fermat's formula and discovered the number to be composite and proved fermat's formula didn't work.
It’s worse than that, even, because it’s believed there are no Fermat primes at all after 65,537. It’s one of the few cases of a conjecture being (with very high probability) 100 percent wrong.
42
42 amusingly appears in 42242 sequences: https://oeis.org/search?q=42&language=english&go=Search
My brain just did not want to accept "7" as the fourth number in the series.
I even drew it, and it was still damned difficult to make my hand offset the line to leave that shit piece in the middle!
Makes me wonder if there's some forth dimension that humans are quantized to ignore. Like right now, there is an alien traffic jam on some sprawling inter-dimensional superhighway caused by a human who plopped down like a duck to check on the price of an NFT.