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Could someone explain this better? The definition in the wiki page appears to be leaving out some information that makes it necessary to understand. Why is 2 +

by casion 3y ago

Could someone explain this better? The definition in the wiki page appears to be leaving out some information that makes it necessary to understand. 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.

LegionMammal978 3y ago
To be fair, the article also says: > 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. [0] https://en.wikipedia.org/wiki/Aliquot_sum https://en.wikipedia.org/wiki/Aliquot_sum
pietroppeter 3y ago
> not expressable the sum of all proper divisor of any positive integer (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
- casion 3y ago
  > In particular the set does not allow for repetition (you cannot count a proper divisor twice). 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.
  - seba_dos1 3y ago
    The information is already included in the definition of untouchable number: > 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.
    - casion 3y ago
      > 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".
      - PeterisP 3y ago
        No, it only assumes that for every integer there exists a single, well defined set of "the proper divisors". Afterwards, summing all of them up is a trivial operation that can only possibly yield a single value.
      - seba_dos1 3y ago
        It says "the sum of all the proper divisors", not "the sum of a sequence consisting of all the proper divisors". It makes no sense to consider repetitions there (it gets easily reduced to absurd when you do), and it's already clear from the definition without even having to look at examples.
  - 3y ago
earthboundkid 3y ago
Math wiki pages are so bad. Would it kill them to use concrete examples? Why does it have to be written so that only mathematicians can understand it? It’s actually not that complicated once you know what they’re talking about, but the entry does nothing to explain it to a layman.
- jiggawatts 3y ago
  It’s a “thing” in certain mathematical circles to reduce everything to the purest possible definition and then refuse to sully that purity with pedestrian nonsense like practical examples. 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.
- svat 3y ago
  The page does have concrete examples in the very first section after the lead? https://en.wikipedia.org/w/index.php?title=Untouchable_number&oldid=1145359635 https://en.wikipedia.org/w/index.php?title=Untouchable_numbe... (avoiding edits from today) — it gives the example of how 4 is not untouchable and why 5 is, and the example it gives of 5 is what the GP is asking about. How would you suggest improving it?
  - earthboundkid 3y ago
    Neither example explains what a proper divisor is.
    - svat 3y ago
      It seems the root comment here had confusion/difficulty not with “proper divisor”, but with “the sum of all the” proper divisors. 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 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.
rendall 3y ago
casion, thanks for asking. I was baffled too by the same things you were, and the replies to your question clarified things.

by casion 3y ago

LegionMammal978 3y ago

To be fair, the article also says: > 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. [0] https://en.wikipedia.org/wiki/Aliquot_sum https://en.wikipedia.org/wiki/Aliquot_sum

pietroppeter 3y ago

> not expressable the sum of all proper divisor of any positive integer (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

casion 3y ago
> In particular the set does not allow for repetition (you cannot count a proper divisor twice). 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.
- seba_dos1 3y ago
  The information is already included in the definition of untouchable number: > 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.
  - casion 3y ago
    > 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".
    - PeterisP 3y ago
      No, it only assumes that for every integer there exists a single, well defined set of "the proper divisors". Afterwards, summing all of them up is a trivial operation that can only possibly yield a single value.
    - seba_dos1 3y ago
      It says "the sum of all the proper divisors", not "the sum of a sequence consisting of all the proper divisors". It makes no sense to consider repetitions there (it gets easily reduced to absurd when you do), and it's already clear from the definition without even having to look at examples.
- 3y ago

earthboundkid 3y ago

Math wiki pages are so bad. Would it kill them to use concrete examples? Why does it have to be written so that only mathematicians can understand it? It’s actually not that complicated once you know what they’re talking about, but the entry does nothing to explain it to a layman.

jiggawatts 3y ago
It’s a “thing” in certain mathematical circles to reduce everything to the purest possible definition and then refuse to sully that purity with pedestrian nonsense like practical examples. 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.
svat 3y ago
The page does have concrete examples in the very first section after the lead? https://en.wikipedia.org/w/index.php?title=Untouchable_number&oldid=1145359635 https://en.wikipedia.org/w/index.php?title=Untouchable_numbe... (avoiding edits from today) — it gives the example of how 4 is not untouchable and why 5 is, and the example it gives of 5 is what the GP is asking about. How would you suggest improving it?
- earthboundkid 3y ago
  Neither example explains what a proper divisor is.
  - svat 3y ago
    It seems the root comment here had confusion/difficulty not with “proper divisor”, but with “the sum of all the” proper divisors. 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 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.

rendall 3y ago

casion, thanks for asking. I was baffled too by the same things you were, and the replies to your question clarified things.