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Untouchable number
- ndsipa_pomu 3y ago[flagged]
- cj 3y agoOther than being theoretically or intellectually interesting, what value do things like “untouchable numbers” have in the real world (or in any practical application)?
- bmacho 3y agoThey help us to develop tools that will allow us faster computation.
- LordShredda 3y agoWell if you click on the link you can see that 5 is the only odd untouchable, much like how 2 is the only even prime. Maybe there's a connection that ties them to cryptography?
- dhosek 3y agoTo be more accurate, 5 i the only known odd untouchable. It’s believed it’s the only odd untouchable, but, like the Goldbach conjecture, it remains likely but unproven.
- dhosek 3y agoNone at this time, but until the advent of modern cryptography, the same was true of primality. Then again, other mathematical curiosities retain their lack of application (and some of us prefer that).
- JadeNB 3y ago> None at this time, but until the advent of modern cryptography, the same was true of primality. 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.
- tantalor 3y agoIf you ever see Paul Erdős mentioned on a math article, it's just for funsies, not real world.
- contravariant 3y agoViewing it as sets lacking a certain property may help explain why it's useful to know and why simply using a countable model is not preferable. 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.
- svat 3y agoYou could ask the same “Other than <its value>, what value does <it> have?” question about anything. (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?)
- permo-w 3y agoyou could ask that, and you would be right to 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
- gizmo686 3y ago> why not numbers that can only be produced by adding 3 primes together? 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. [0] https://arxiv.org/pdf/1501.05438.pdf https://arxiv.org/pdf/1501.05438.pdf
- permo-w 3y agothe actual examples I give are just that, examples. why not numbers that can only be produced as the sum of 17 primes? or 459? or numbers that have the same number of factors as their digits added together does? there are infinite of these constraints that can be invented. why is this one particularly interesting
- raincole 3y agoMathematicians are almost always working on things without immediate applications. Because if they work on something that has immediate applications, they'd be called computer scientists, physicists, statisticians, etc.
- amelius 3y agoYeah, without application this sounds almost as silly as: https://en.wikipedia.org/wiki/Numerology https://en.wikipedia.org/wiki/Numerology
- earthboundkid 3y agoThis is the opposite of numerology. Numerology says that numbers have practical meaning, like 7 is lucky or 13 is unlucky. The point of pure math is that it has no practical meaning. We study it because we are free and not slaves.
- amelius 3y agoIt can be opposite but still as silly.
- svat 3y agoIt is numerology, with impressive pedigree: sum-of-divisors and “perfect numbers” are in Euclid: https://en.wikipedia.org/w/index.php?title=Perfect_number&oldid=1166654700#History https://en.wikipedia.org/w/index.php?title=Perfect_number&ol... (for the numerology connection, search the page for "created in 6 days" and for "numerology"). See also https://en.wikipedia.org/w/index.php?title=William_of_Auberive&oldid=1069314286 https://en.wikipedia.org/w/index.php?title=William_of_Auberi... , and https://en.wikipedia.org/w/index.php?title=Aliquot_sum&oldid=1172322505 https://en.wikipedia.org/w/index.php?title=Aliquot_sum&oldid... which mentions: > The mathematicians Pollack & Pomerance (2016) noted that one of Erdős' "favorite subjects of investigation" was the aliquot sum function.
- xyst 3y agoMaybe as a seed generator
- lubujackson 3y agoReminds me of the number my son invented when he was 4. A killion. It's a number "so big, ya die."
- Lichtso 3y agoNot as a number, but as a unit it actually exists. It is called a "mort" (from mortality). In that sense one mort is "so much, you'll die". Though, the commonly used scale is a mort * 10 ^ -6. https://en.wikipedia.org/wiki/Micromort https://en.wikipedia.org/wiki/Micromort
- seeknotfind 3y agoIt takes 9 million micromorts to kill the average cat.
- kccqzy 3y agoA great opportunity to begin teaching your son some set theory until he understands inaccessible cardinals!
- Arcorann 3y agoFunnily enough, there was a short story about the killion published in the New Yorker in 1982: "The killion, as every mathematician knows, is a number so big it can kill you." [1] [1] https://www.newyorker.com/magazine/1982/09/06/the-killion https://www.newyorker.com/magazine/1982/09/06/the-killion
- xNeil 3y agoReincarnation, maybe?
- pharrington 3y agoa true mathematician in the making
- tnecniv 3y agoWait until he learns about “Killing fields” which are vector fields whose flow preserves distance and named after Wilhelm Killing
- casion 3y agoCould 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 agoTo 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.