Long ago, I had an idea of creating (actually, labeling) a clock that will have, instead of the numbers 1,…,12, important mathematical problems whose solution turned out to be that number. Such a clock could be awesome for mathematics departments.
This is a call for such problems. Please, consider posting problems for numbers that are not well-addressed yet.
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I think I have a good source for you, provided you are looking for wordy yet mathematically interesting properties. The wikipedia pages for the integers from 1 to 12, entitled 'n_(number)', all have interesting mathematical facts about those numbers in the "Mathematics" section. For instance, 7:
- Seven is the lowest natural number that cannot be represented as the sum of the squares of three integers.
- n = 7 is the first natural number for which the next statement does not hold: "Two nilpotent endomorphisms from Cn with the same minimal polynomial and the same rank are similar."
- 7 is the only dimension, besides the familiar 3, in which a vector cross product can be defined.
- 7 is the lowest dimension of a known exotic sphere, although there may exist as yet unknown exotic smooth structures on the 4-dimensional sphere.
- etc.
These have some your numbers in them...
What do you get when you link primes, transcendentals, $0$, unity, subtraction, and equality? $$e^{2 \pi i}-1=0$$ A root inside a root inside a root... $$2=\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}}$$
What do you get when you sum all the natural numbers? Not what you think according to the Riemann zeta function. \begin{equation} 1+2+3+4...={-1 \over 12} \end{equation}