Find integers a, b, c, such that a³ + b³ + c³ = 42

The solution for this problem was found recently (2019):

Here is an almost example, it can be solved, but its hard: https://www.quora.com/How-do-you-find-the-positive-integer-solutions-to-frac-x-y+z-+-frac-y-z+x-+-frac-z-x+y-4/answer/Alon-Amit

$n=\lfloor \sin (3\uparrow \uparrow \uparrow \uparrow 3)\rfloor$

Not sure if that is counter to the spirit of the post, because while the answer is either -1 or 0, it involves an intermediate calculation which vastly exceeds ${10}^{100}$.

Your rules need refining about how large "intermediate values" are:

⌊π^π^π^π^π⌋ mod 10

-High school formula

-Integer result < 10 < 10^100

-Can't be solved w/ contemporary maths

-Misses the spirit of the challenge but obeys the rules

What is the smallest positive integer n such that both p and p+n are prime for infinitely many values of p?

(The answer is probably 2, but so far we can only prove that it is not greater than 246.)

Source.

Maybe anything about Hofstadter's Q-sequence.

In the Q-sequence, $Q\left(n\right)$ is defined as $Q(n-Q(n-1\left)\right)+Q(n-Q(n-2\left)\right)$ (with terms n=1 and n=2 set to 1). So $Q\left(3\right)=Q(3-1)+Q(3-1)=2$, and so on. This sequence "dies" if $Q\left(n\right)>n$ for any $n$, because then the next term would ask for some term like $Q(-1)$ which is undefined. We don't know whether the Q-sequence dies or not.

There's a Math Stack Exchange question: "Conjectures that have been disproved with extremely large counterexamples?" Maybe some of the examples in the answers over there would count? For example, there's Euler's sum of powers conjecture, which only has large counterexamples (for high k), found via ~brute force search.

I'm kind of annoyed by the challenge, also.

However, this does provide the opportunity to think a bit more deeply about identity and being able to refer to a thing, like π.

To expand on this, the number n, like the number π, has no decimal representation, but unlike most real numbers, it is well-defined since it can be pointed to and thus named: the equation as given in the post is one such name (it is an implicit representation of n; simple algebra may be used to rewrite it into a more explicit form).

If n became useful enough, we could even give it a simpler name (like we did with π and 2), instead of the compound symbolic expression.

There are a lot of unspecified assumptions in Ender's challenge, such as reducing the expression to some canonical form, e.g., 2 instead of 16/8, sqrt(4), or the only even prime number. It takes a bit of sophistication to appreciate the depth of questions of equality, identity, and representation.

The fact that Ender had not considered any of this while confidently presenting the challenge leaves me feeling a bit annoyed.