[HN Gopher] Untouchable number
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Untouchable number
Author : optimalsolver
Score : 71 points
Date : 2023-08-25 14:37 UTC (8 hours ago)
(HTM) web link (en.wikipedia.org)
(TXT) w3m dump (en.wikipedia.org)
| cj wrote:
| Other than being theoretically or intellectually interesting,
| what value do things like "untouchable numbers" have in the real
| world (or in any practical application)?
| svat wrote:
| You 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 wrote:
| you 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 wrote:
| > 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
| permo-w wrote:
| the 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
| contravariant wrote:
| Viewing 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.
| dhosek wrote:
| None 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 wrote:
| > 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 wrote:
| If you ever see Paul Erdos mentioned on a math article, it's
| just for funsies, not real world.
| deepspace wrote:
| What does that even mean?
| effie wrote:
| Paul Erdos is known for work in hobby mathematics which has
| little or no use "in real world".
| LordShredda wrote:
| Well 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 wrote:
| To 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.
| bmacho wrote:
| They help us to develop tools that will allow us faster
| computation.
| lubujackson wrote:
| Reminds me of the number my son invented when he was 4. A
| killion. It's a number "so big, ya die."
| kccqzy wrote:
| A great opportunity to begin teaching your son some set theory
| until he understands inaccessible cardinals!
| Lichtso wrote:
| Not 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
| ndsipa_pomu wrote:
| [flagged]
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