I couldn't think of anything that matches this topic, so decided to just post a moe picture. Say hello to the chocolate cornet.

It's a puzzling fact that sometimes certain little ingredients of mathematics (by which I mean a simple equation, or even just a few numbers) just pop up in different, possibly totally (seemingly) unrelated areas in ways one cannot easily imagine.

Two layman's examples are $e$ and $\pi$--they're freaking everywhere. A slightly less layman example is Euler's constant: the wiki article even has a section summarizing its various appearances. A much less layman example is the connection between (jargon alert) irreducible representations of the monster group and the Fourier expansion of the j-invariant modular function, which later led to the moonshine theory. Another example I met recently was the classification of simple Lie algebrae, which ended up using a result that I learned in grade school: that there are only very few distinct integral solutions to the equation $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} > 1$.

Sorry for piling up examples before even getting into my main topic. It's just my personal hobby to play with those examples; not only are they cool, but they also often give me better understanding of what on earth I'm working on. In a subject such as mathematics, you really kind of need that intuition.

The new one I noticed today is the number pair 3 and 4. Quite routine numbers? Well, let me list some of the cases where I have met them:

• The volume formula of a sphere, $V = \frac{4}{3} \pi r^3$. (I learned this in gradeschool)
• Triple-angle formula for trignometry, $\sin(3x) = 3 \sin(x) - 4 \sin^3(x)$. (I learned this in middle school)
• The smart tweak used in the determinization proof of Toda's theorem (Lemma 4.3), $x \mapsto 3x^4 + 4x^3$, which is used to aggregate parity-SAT instances. (I learned this in high school)
• Polynomial behavior modulo a composite number, $x \equiv (4x+3)(3x+4) \pmod{3 \cdot 4}$. (I learned this today)
• $3^4 - 4^3$ is the (3 + 4)th prime number. (I realized this two minutes ago)

Some really beautiful stuff there, isn't it? Well, aside from appreciating the magical recurrence of this 3-4 pair, let's also spend some time thinking why this is the case; that is, why do they show up? Any ways to draw connections among them?

Funny thing is, I can't seem to think of much. The first one is pretty straightforward; the number 3 there is merely the dimension of the (local, approximate) Euclidean space that we live in, and the 4 comes from the fact that $1 + \frac{1}{3} = \frac{4}{3}$. But for the second one, I can't really think of any "cool" explanations beyond trigonometric manipulation. Toda's trick is really a smart little piece, but as far as I can think, no much can be said about it with respect to the other formulae that I listed above. (If you have some good ideas, please let me know via comments.)

The polynomial one, on the other hand, is actually a lie; as the reader can readily verify, this formula works for all pairs (n, n+1). It's worth mentioning that the factorization is not irreducible when modulo 12; in the paper the author used modulo 6, in which case the factorization is indeed irreducible. Well, then of course comes the natural question: can this factorization ever be irreducible for some n? It seems to me the answer is negative, and it should probably not be too hard to prove. I'll get back onto this later. (If you find a proof, please comment.)

The last one is of course again worth some playing with. The natural question to ask is, what kinds of n can make $n^{n+1} - (n+1)^{n}$ a prime number? A simple computer computation (based on Miller-Rabin algorithm) shows that solutions when n is less than 1000 are 3, 6, 9, 12, 44 and 883. Of course, the existence of 883 (which I didn't think would pop out) suggest that there might be infinitely many of them, but my number theory skills are way too little for me to even try to approach this problem. Maybe in the future we'll revisit this little puzzle. (Update: some additional interesting properties about this sequence can be found here.)

That's quite some diversion...but really the problem is that I don't have much to say about those 3-4 pairs. It seems the reason they appear in such nice forms is simply that they are really, really small. And as we know well from group theory, small things tend to have strange properties. (Quantum physicists would probably agree.) Well, regardless of the reasons behind them, those little formulae and numbers are still quite fun to play with, are they not?