5 ms·

Other than being theoretically or intellectually interesting, what value do things like “untouchable numbers” have in the real world (or in any practical applic

by cj 3y ago

Other 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 ago
They help us to develop tools that will allow us faster computation.
LordShredda 3y ago
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 3y ago
  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.
dhosek 3y ago
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 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 ago
If you ever see Paul Erdős mentioned on a math article, it's just for funsies, not real world.
- deepspace 3y ago
  What does that even mean?
  - effie 3y ago
    Paul Erdos is known for work in hobby mathematics which has little or no use "in real world".
contravariant 3y ago
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.
svat 3y ago
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 3y ago
  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 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 ago
      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
raincole 3y ago
Mathematicians 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 ago
Yeah, without application this sounds almost as silly as: https://en.wikipedia.org/wiki/Numerology https://en.wikipedia.org/wiki/Numerology
- earthboundkid 3y ago
  This 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 ago
    It can be opposite but still as silly.
- svat 3y ago
  It 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 ago
Maybe as a seed generator

by cj 3y ago

Other 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 ago

They help us to develop tools that will allow us faster computation.

LordShredda 3y ago

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 3y ago
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.

dhosek 3y ago

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 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 ago

If you ever see Paul Erdős mentioned on a math article, it's just for funsies, not real world.

deepspace 3y ago
What does that even mean?
- effie 3y ago
  Paul Erdos is known for work in hobby mathematics which has little or no use "in real world".

contravariant 3y ago

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.

svat 3y ago

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 3y ago
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 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 ago
    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

raincole 3y ago

Mathematicians 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 ago

Yeah, without application this sounds almost as silly as: https://en.wikipedia.org/wiki/Numerology https://en.wikipedia.org/wiki/Numerology

earthboundkid 3y ago
This 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 ago
  It can be opposite but still as silly.
svat 3y ago
It 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 ago

Maybe as a seed generator