What’s the least whole number which cannot be defined by a phrase of less than a twenty English words?
The Least Whole Number
In direct response to the prompt, this isn’t specific enough. I assume “the least whole number which cannot be defined by a phrase consisting of less than a hundred words” is not a suitable answer. Are (any) axioms assumed? Does the number have to be the “result” of a “nontrivial” computable function? (e.g. a function could be defined constantly as Rayo’s number, but Rayo’s function is noncomputable. I know this is pretty vague)
If anything goes I submit my entry as two words: Rayo’s number. Which might be cheating but the actual definition of Rayo’s number doesn’t exceed 100 words either. Which still might be cheating. Rayo’s number is in short the result of a similar question being posed and two professors fighting over it.
Also see here for some more interesting reading on large numbers (if you haven’t already), and the inspiration for the contest which led to Rayo’s number.
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Very interesting! I like Scott Aarsonson’s blog and have one of his books. The blog post you linked actually references an equivalent version of this riddle! To make it easier I’ve now reduced the problem to 20 words instead of 100.
If you haven’t seen it, you might like this. One of his students (and he) made a Turing machine that ZFC cannot prove halts. They also made some that will halt iff there’s a counterexample to the Riemann Hypothesis.
I think the most obvious response is that if such a number existed, I would be able to say “the least whole number that cannot be described in 20 words or fewer,” and therefore, no such number could exist.
The other route to take would be to go with the examples Nick sent, which rely on common understanding and being reasonable. It would shock me if that’s what you were after, as those are ideas you have never shown interest in before.
If we are to go down that route, my initial thought is the idiotic “111111111111111111111” which can obviously be described as 21 ones. What we need, then is something with a good amount of randomness. I also think that if it had any factors, you could likely describe it more concisely by referring to the factors. What the number is, then, is the smallest 21-digit prime number with no more than 2 consecutive identical digits.
While this number is defined, no person could reasonably be able to figure out what it was, unless the full 21 digits were read out to them.
I like your idea of primes, although you’d want to be sure the prime number cannot be described as something like “the eleventh prime number with more than 20 digits”.
This is really what I’m going for. 
But how can no such number exist?? There are a finite number of phrases with less than 20 words. Most map to no number, of course, and some might be ambiguous and thus inadmissible. But certainly there are numbers which are not mapped to, under any interpretation. And of those, of course there is a smallest.