Other than being theoretically or intellectually interesting, what value do things like “untouchable numbers” have in the real world (or in any practical application)?
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.
Because if they work on something that has immediate applications, they'd be called computer scientists, physicists, statisticians, etc.
(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.
> The mathematicians Pollack & Pomerance (2016) noted that one of Erdős' "favorite subjects of investigation" was the aliquot sum function.
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.