Why the Perfect Cuboid Problem Remains Unsolved – Math History

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Why hasn't Perfect Cuboid Problem been solved yet, whereas (possibly) more nontrivial ones such as FLT and Sphere packing have been solved?

I understand that calling some problems more nontrivial may be naive and seemingly trivial problems can be deceptively tricky, as with the FLT. All the same, FLT and to a lesser extent, Sphere Packing, garnered lots of attention by successive generations of mathematicians, until someone decided to finish it off and succeeded.

But, AFAIK, the Perfect Cuboid (PC) problem hasn't generated this kind of attention, perhaps because Fermat didn't leave a note about it. Is that the reason for PC remaining unsolved? One of the standard references for PC , Unsolved Problems in Number Theory , suggests several numerical results (p.178), but of course nothing like a proof, much like the status of FLT and Sphere Packing many decades ago.

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If I were to take a guess, I'd suggest that the reason it hasn't been solved yet is because there's not any apparent practical application, and nobody's put a bounty on it that's large enough to make it worth anybody's trouble to solve.

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Information Theory – Has Error Correction Been Solved?

This is largely true in the sense that long LDPC codes are within a fraction of a dB of the Shannon limit in AWGN channel. Their design (by a process called density evolution - don't look at me, I don't know the math) also seems to be relatively simple. It depends on stochastic math as opposed to algebra/combinatorics of finite fields/rings.

My limited experience only allows me to make the following remarks:

The block length of an LDPC code needs to be relatively large. This is one of the reasons they were not chosen to LTE standard (European/Asian 4G cellular standard). My understanding about the `best' code in a radio channel is (as a function of the block length): less than 100 bits => use an algebraic block code, 40-400 bits => use a convolutional code, 300-20000 bits => use a turbo code, 5000 bits+ => use an LDPC code (the intervals overlap as I am not willing to bet the family fortune on any one of them). IIRC this is how it is done in LTE (release 9), except that in that cellular standard the longest data block has 6144 bits (so handled with a turbo code).
The design of an LDPC code needs to take into account that it is usually not practical to base it on a random Tanner graph with the prescribed weight distribution. In particular, when it is supposed to be used on a handheld device. The routing problem of the belief propagation algorithm becomes prohibitive otherwise. So the LDPC codes specified for radio communication standards a few years back (DVB-S/T2 in Europe, Mediaflo in the US) were designed around a particular decoding circuitry, or rather the Tanner graphs had to come from a certain kind of a family. I have never been truly up to speed with these aspects, so some progress may have been made.
When higher order modulation is used there are further problems and considerations that need to be taken into account. It is not entirely obvious that we should stick to LDPC codes based on bits alone in that case, because the channel will create dependencies among the LLRs of individual bits. But the BP algorithm becomes very complicated, if we want to feed something other than LLRs of individual bits to it.
Getting rid of the error floors is still a challenge. Alas, I'm not up to speed with that either. Possible workarounds involve using catenated code with a light weight algebraic outer code handling the residual errors.

Caveat: I have only studied this cursorily (enough to implement an LDPC decoder and simulate/analyze some of the codes). These are recollections and impressions rather than hard facts. You can get the minutes of the relevant meetings of workgroup in charge of this within LTE from the web pages of 3GPP (=third generation partnership project).

Other types of codes will remain to be useful for specific purposes. There is no substitute to RS (or AG, if you need longer blocks), when you only have hard byte errors. LDPC (and Turbo) codes are designed with soft bit errors (=reliability estimates of received bits) in mind. This is a valid assumption in all wireless communication. Also network coding appears to be a hot topic currently. There the code operates at the level of data packets that may or may not arrive at all. Also look up Raptor code, if this type of problems are of interest to you.

Algebraic coding theory is not dead yet, but it does concentrate on problems other than designing ECCs for bulk data transmission. An algebraic machinery yielding LDPC code designs guaranteed to have very low error floors (= high minimum Hamming distance) may be out there, but I don't know what it would be.

[Math] Why has $\int \sin (\sin x) dx$ not been solved yet

Well you can do

$$\int_0^{\pi}{\sin(\sin x)} = \pi H_0(1)\,,$$ where $H$ is a non-elementary function known as the Struve function. (For info on Struve functions, see http://mathworld.wolfram.com/StruveFunction.html.)

You might be wondering...what good does it do to express the integral in terms of a function that almost nobody has ever heard of?

Well, think back to a long time ago when the sine, cosine, and tangent functions themselves were as arcane as Struve functions are now. A math student might have asked a question analogous to your own, saying "Why hasn't anybody been able to integrate $\int{{dx\over 1+x^2}}$? The specialists at the time would have said, "Actually, we ${\it can}$ evaluate that integral:

$$\int{{dx\over 1+x^2}} = \arctan x\,.$$

If the student were to say, "What good does that do?" The expert would have said, well, this is helpful because see we have these books with values of the tangent function written down in them. So effectively we have "done the integral" because we know which book to look in.

Likewise today, it helps to express the integral $\int_0^{\pi}{\sin(\sin x)}$ as $\pi H_0(1)$ because now we know which book to look at. (Probably Abramowitz and Stegun's ${\it Handbook\ of\ Mathematical\ Functions}$, which I used to use a lot when I was a mathematical physicist.) More commonly, programs like Mathematica have all those values pre-loaded, along with approximation schemes for computing accurate values by interpolation.

And you know, that's all your calculator is doing when it computes $\sin x$ anyway. Once you're beyond the polynomial functions, you're pretty much in this territory.

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Information Theory – Has Error Correction Been Solved?

[Math] Why has $\int \sin (\sin x) dx$ not been solved yet

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