[tzs]

It's nice to see something other than e^(ᴨi)=-1.

If I had to get a math tattoo, I think I'd go for lim n→∞ Q_n^(1/n) = e^(ᴨ^2/(12 log 2)).

That comes from a theorem proved by Khinchin and Lévy. Khinchin proved that for almost all real numbers if you take the sequence of convergents of their continued fraction expansion, {P_1/Q_1, P_2/Q_2, ...}, then the sequence {Q_1, Q_2^(1/2), Q_3^(1/3), ...} approaches a limit, which is the same limit for almost all real numbers. Then Lévy determined the value of that limit, which is now called either Lévy's constant or the Khinchin–Lévy constant.

If not that, then this (in standard math notation rather than the verbose notation I'm using here):

Line 1: Let H_n = sum i=1 to n 1/n

Line 2: Hypothesis: sum d|n d < H_n + e^H_n log(H_n) for all n > 1

That's neat because that hypothesis is true if and only if the Riemann hypothesis [1] is true [2].

The Riemann hypothesis is a conjecture about complex numbers, and is widely considered to be the most important unsolved problem in pure mathematics. That it turns out to be equivalent to a such a simple conjecture involving just integers and a couple real functions from pre-calculus is a surprise.

[1] https://en.wikipedia.org/wiki/Riemann_hypothesis

[2] https://arxiv.org/abs/math/0008177

[BlueTemplar]

I'm curious about that "almost all" ?

[SamReidHughes]

All but a set with measure zero, in this case.

[BlueTemplar]

Why just not say "non-zero" then ?

[tzs]

"non-zero" is not the same as a "all but a set of measure zero". Here's what "measure zero" means:

A set S of real numbers has measure zero if for any positive ε no matter how small, there exists a countable set of intervals such that (1) every element of S is in at least one of the intervals, and (2) the total length of the intervals is < ε.

For example, let S be the set of positive integers, {1, 2, 3, ...}. Proof: consider the set of intervals {I_1, I_2, I_3, ...}, where I_n is the interval [n-ε/2^(n+2), n+ε/2^(n+2)]. Every member of S is contained in one of these intervals.

The length of I_n is ε/2^(n+1). The length of all the intervals is ε(1/4 + 1/8 + 1/16 + ...) = ε/2 which is < ε.

Thus S, the set of positive integers, has measure zero.

A similar argument works for any countable set of real numbers, such as the rational numbers or the algebraic numbers, and so something that was true everywhere except at rational numbers would by true for "almost all" real numbers.