Untouchable number
en.wikipedia.org
en.wikipedia.org
Though, the commonly used scale is a mort * 10 ^ -6. https://en.wikipedia.org/wiki/Micromort
[1] https://www.newyorker.com/magazine/1982/09/06/the-killion
Why is 2 + 2 + 1 = 5 not sufficient? It doesn't say unique proper divisors. The definition of proper divisors doesn't seem to explain either.
i.e. why is 10 not an untouchable number but 5 is?
Specifically given the precise definition given, anything should be touchable as 1 is a proper divisor, and you can sum any number of 1s to "touch" a number. Clearly we're missing some implicit restriction.
> That is, these numbers are not in the image of the aliquot sum [0] function.
And that function is specifically defined, using ∑ notation, as the sum of all (distinct) integers that are the proper divisors of its input.
More generally, in my experience, uniqueness is generally assumed when talking about divisor sums and their variants, unless otherwise indicated.
(Emphasis on all mine)
A proper divisor is a positive integer divisor of n other than n. Examples: 1 is a proper divisor of all positive integers except 1, 2 is a proper divisor of all even integers except for 2, 3 is a proper divisor of 6, 9, ..
By all proper divisor of a specific positive integer n we mean the set of all positive integers that divide n and are less than n. In particular the set does not allow for repetition (you cannot count a proper divisor twice). So 1+2+2=5 is not valid since you are counting twice 2.
10 is not untouchable since 1,2,7 are all proper divisor of 14.
5 is untouchable because it cannot be 1+p+q with p < q since p > 1 so q > 3 so 1+p+q>5 (recall all proper divisor are distinct). It cannot also be 1+p because then p=4 and if 4 is a proper divisor also 2 is a proper divisor of a number so 1,4 is not the set of all proper divisor of any number
How would I know this having read the description/definition?
I checked multiple definitions for proper divisor and untouchable number before I wrote my post and I could not find anything explicit.
Thank you for the explanation btw. Still a bit hung up on how I could have figured that myself given the information presented.
> a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer
There's only one sum of all the proper divisors for any given integer.
This really seems to assume that uniqueness is an implicit property of each integer of such sums. I don't understand how you would know know that or how to discover that other than "you couldn't get the answers we're showing you unless you assumed that".
> The number 4 is not untouchable as it is equal to the sum of the proper divisors of 9: 1 + 3 = 4.
If repeated divisors were counted, that sum would be 7, not 4.
Another way to realize it would be to notice that 1 is a very special case if repetition counted. Would 1 be represented as a proper divisor infinite times? that would make every sum infinite. Or would there be a special case for why 1 is only counted once, but other numbers can be counted multiple times? or do you just exclude 1 from being a proper divisor?
If all the sums were infinite it wouldn't be interesting.. I suspect with the other 2 types of rules, almost none of the numbers would be "untouchable", which also makes it not interesting.
Count 1 only once and allowing repetition is this way. I found that 2 is "untouchable" since sum(1) = 1, sum(2) = 3, sum (3) = 4, sum(4) = 5. It's easy to see that every other sum will be larger than two.
All other even number is "touchable": for even n, take the lowest prime less than n and then multiply it by 2 a few times. With the 2s and the odd prime and the extra 1, it can be equal to n.
Similarly all other odd numbers are "touchable": for odd n, take the lowest prime p less than n and either multiply it by 2 if p = n - 2, or multiply it by 3 to make it even, then multiply by 2 one or more times and with the 1 it becomes odd n.
I suspect excluding 1 in the sum but allowing repetition is similarly trivial.
Anyway, yes, reading such definitions comes with experience (what's called “mathematical maturity”: https://en.wikipedia.org/wiki/Mathematical_maturity) — you have to assume that everything being stated is well-defined and makes sense, that every word matters (e.g. you can't ignore the word “all” in “the sum of all proper divisors”, nor can you ignore the word “the”!), and then, if that doesn't work, try to see if alternative definitions make sense. Also, sometimes things are stated in two different ways; that's usually helpful. In this case, given that the first two sentences are:
> An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer (including the untouchable number itself). That is, these numbers are not in the image of the aliquot sum function.
Note the “That is”: If the first way of stating the definition doesn't seem clear / doesn't seem to fit the example that immediately follows, then try the second sentence, specifically follow the “aliquot sum” link to https://en.wikipedia.org/w/index.php?title=Aliquot_sum&oldid... which makes it very clear:
> For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are 1, 2, 3, 4, and 6, so the aliquot sum of 12 is 16 i.e. (1 + 2 + 3 + 4 + 6).
[…]
> The untouchable numbers are the numbers that are not the aliquot sum of any other number.
Any attempt at requesting clarification is met with: “This is the only fully general definition” or some such.
It inevitably leads to articles that can’t be understood even in principle without understanding everything else already, because simple concepts are rephrased in terms of the most general (most abstracted) concepts.
My favourite is that they never miss an opportunity to rephrase alternation like 0,1,0,1 in terms of exponentials raised to complex powers.
Or computer algorithms that couldn’t have existed before the nineteen hundreds using symbols from Ancient Greek and maybe two other character sets just to make it more spicy if you want to “translate” it back into mere code.
How would you suggest improving it?
In any case, if your problem is that this page doesn't explain "proper divisor", then note that the phrase "proper divisor" in the first sentence is a link that goes to the relevant section of the [[divisor]] article, which has lots of examples.
If the complaint behind “Math wiki pages are so bad” is simply that not every page explains everything from scratch but relies on the reader having to follow links, then this is an inherent property of a random-access reference work like an encyclopedia (rather than a careful linear presentation like a textbook), and the fact that mathematics is a subject with quite some depth (where understanding a topic requires understanding several others first).
This is not unique to mathematics articles, e.g. if you go to the Wikipedia article on "init", it says:
> In Unix-based computer operating systems, init (short for initialization) is the first process started during booting of the operating system.
where "Unix", "operating system", "process" and "booting" are wiki links: you need to follow the links if necessary and understand them first, as this page won't start by explaining what a computer is, what operating systems are, etc. This is true for basically all topics: clicking on https://en.wikipedia.org/wiki/Special:Random a few times, I find:
> Ustilaginoidea is a [[genus]] of [[fungi]] in the family [[Clavicipitaceae]].
where you need to understand "genus" and "fungi" first, or
> Masaumi Shimizu is a Japanese former professional baseball Catcher,and current the fourth squad battery coach for the Fukuoka SoftBank Hawks of Nippon Professional Baseball (NPB).
where "catcher", "battery" etc are links that need to be followed (this page won't start from first principles and explain sport, baseball, catcher, etc).
I think it may be instructive to compare Wikipedia math pages to something professionally published and edited like, say, The Princeton Companion to Mathematics. I just did that for a few random topics, and Wikipedia was in some cases easier to read and in some cases harder: it was not consistently better or worse. But doing this for more pages may be instructive — or simply pick some random math pages and show how they can be improved, while still remembering that in an encyclopedia much of the information necessary to understand a certain page will inevitably be at other pages.
Untouchable numbers are a very simple idea. Any elementary school student who knows about divisors and prime numbers can have it explained to them in about ten minutes. The Wikipedia page for the article should be pitched so that a layman can understand it.
I understand that if you’re writing a cookbook, you have to decide if you’re pitching it first time chefs who need “how to boil water” explained to them or professional chefs who just need references for some ratios. What serves one audience wastes the time of the other. If you want to know how init works, you probably already know what Unix is. If you’re looking up a particular species of mushrooms, you probably know how genii and species work.
Untouchable numbers is a simple concept. It should not be pitched at professionals first. It should start with an explanation that a middle schooler can understand and then move on to advanced explanations in later sections.
• That there are no concrete examples on the page: this is not true, the page has a couple of concrete examples (the numbers 4 and 5), and I added a third one (the number 6) earlier today.
• That “proper divisor” and “sum of all the proper divisors” aren't explained: but these are explained on the pages (divisor and aliquot sum respectively) linked from the first two sentences of the article, in keeping with the nature of an encyclopedia.
So what is the problem exactly? What explanation/wording would you suggest? It may be easier to understand the general statement “Math wiki pages are so bad”, if there was an answer to “How would you suggest improving it?” for this specific wiki page.
[I have a theory, that this has to do with “depth”: to understand this concept one needs to understand, in order, (1) the idea of divisors, (2) the definition of a proper divisor of N as any divisor of N other than N itself, (3) the idea of the sum of all the proper divisors of N, (4) finally, the idea of an untouchable number, as a number not achievable as such a sum for any N. Each step is individually easy, but because of this ordering requirement, overall there is “depth”. In fact I actually think that any middle-schooler would have no trouble learning the concept from Wikipedia as it stands, if they read the relevant sentences from the relevant Wikipedia articles in order, e.g. the following from https://en.wikipedia.org/wiki/Aliquot_sum seems clear enough for (3):
> For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are 1, 2, 3, 4, and 6, so the aliquot sum of 12 is 16 i.e. (1 + 2 + 3 + 4 + 6).
It's just that no single page explains all four ideas in order. So the middle-schooler doesn't need to be an experienced mathematician, just experienced at reading Wikipedia and following links. (Arguably this is part of mathematical maturity, i.e. understanding definitions as needed until you can understand the original definition.)
The process you describe: “I eventually figured it out on my own by, yes, examining the two examples they gave and proving to myself how it works” seems to suggest things are working as intended; it's just the nature of mathematical concepts that they require a little bit of thought! As you say, “Any elementary school student who knows about divisors and prime numbers can have it explained to them in about ten minutes.” (I agree, and one doesn't even need prime numbers): about 5–10 minutes is how long it takes, but readers complain about mathematical articles because it's uncomfortable that reading a handful of sentences should require several minutes to understand, when actually this is inevitable. This reminds me of the “monad tutorial fallacy” https://byorgey.wordpress.com/2009/01/12/abstraction-intuiti... — after a bit of struggle to understand, when it eventually clicks, the reader says “ah it's so simple” and thinks that the earlier explanations were bad. But that's just my theory, and if you have a concrete suggestion for how the article could be improved, that may be revealing.]
I'm not sure that this is true, at least if you are flexible about what counts as an ‘application’. The concept of divisibility, and then of primality, surely developed from considerations of how a certain number of objects could, or could not, be broken into groups, say for storage or transport. To know that there are several ways to group 24 objects, but only two (trivial) ways to group 23 objects, is an application, even if it's not especially sophisticated.
Uncountability means that real numbers lack certain properties. If you accept the claim of physicists that the world is best described using real numbers then this has some applications.
Among the things that are impossible are things like constructing a function to pick a number for each set of real numbers. Or making an algorithm to decide two numbers are equal.
Even more concretely the fact that it is incredibly hard to determine whether something is non-zero (or even nonnegative) is the bane of various numerical algorithms. Obviously you can work around these issues, but uncountability is the first sign of trouble.
(The answer is: none. For that matter, how would you answer questions like: what value does the concept of even-and-odd numbers have? Or, say, Fibonacci numbers: sure the Fibonacci sequence itself might have some applications, but what value does knowing whether or not a certain number is a Fibonacci number have?)
untouchable numbers have what seems like a pretty arbitrary definition and yet the article mentions the numbers being studied a thousand years ago, which begs the question: why? why not numbers that can only be produced by adding 3 primes together? why not only numbers that can be produced by multiplying squares greater than 1? there are infinite unique infinite sets of integers. why is this one more interesting than the other infinity to the degree that it's been studied for a thousand years?
if it's given that the Fibonacci sequence has uses, then knowing the numbers in that sequence is also obviously going to be useful
Goldbach's weak conjecture: Every odd number greater than 5 can be expressed as the sum of three primes.
First proposed in 1742, and proven in 2013 [0]. The original proposal considered even numbers as well, nowadays those are covered by Goldbach's strong conjecture, with a tighter bound of 2 primes.
> why not only numbers that can be produced by multiplying squares greater than 1?
You mean squares containing at least 2 distinct prime factors? Fully classifying this set of integers would fit well on an undergrad intro to proofs exam.
learning these things brings greater understanding and greater understanding brings advances in multiple fields.
the most fundamental questions are going to be answered (or at least conceived of and worked on) first. answering the most fundamental questions often brings the largest leaps in understanding as well, which is why untouchable numbers are considered more notable than "numbers which can be expressed as the sum of 17 primes", and so on.
Because if they work on something that has immediate applications, they'd be called computer scientists, physicists, statisticians, etc.
> The mathematicians Pollack & Pomerance (2016) noted that one of Erdős' "favorite subjects of investigation" was the aliquot sum function.