Oops! Yes, of course.
Problems like
are loosely related to Waring’s Problem: Can all (sufficiently large) integers be expressed as the sum of n k’th powers?
For example, it is conjectured that all integers larger than 7.4 trillion can be expressed as the sum of four non-negative cubes.
While not as famous as Fermat’s Last Theorem, Waring’s Problem has attracted huge interest; proofs for various cases are associated with the names of some of the greatest mathematicians.
This is great! This is what I was thinking of! I would have never found that on my own.
Interesting that there are solutions for ‘lesser’ integers. The example given for the 5th power:
27[sup]5[/sup] + 84[sup]5[/sup] + 110[sup]5[/sup] + 133[sup]5[/sup] = 144[sup]5[/sup]
means there is a solution with only 4 variables instead of 5. So there is no pattern of increasing variables matching increasing exponents like I was imagining. So in the case of FLT, if I am understanding this correctly, is that for a[sup]n[/sup] + b[sup]n [/sup] = c[sup] n [/sup], there is no ‘lesser’ power answer (for n=3, for example), which invalidates the progression I was imagining, which is why FLT is so important, I gather. Neat!
Itself, KarlGauss, Indestinguishable, and anyone I’ve missed, thanks for the replies.
I don’t have it off the top of my head (unsurprisingly). I do recall reading a search on the sum of five fifth powers equalling another fifth power. Two solutions discovered had 0[sup]5[/sup] as a one of the terms. In other words, there are (at least) two known instances where the sum of four fifth powers equals another fifth power.
that is, a[sup]5[/sup] + b[sup]5[/sup] + c[sup]5[/sup] + d[sup]5[/sup] = e[sup]5[/sup]
That, to me is interesting. When I see such things I always want to know if there is a generalised pattern or some way of finding/predicting further similar incidents.
That’s what got me started on this. I wanted a pattern, and it seemed that FLT was going against the ‘pattern’. That it, if every increase in exponent required a corresponding increase in number of variables, then it was silly to assume that there would be solutions for ‘number of exponent minus 1’ number of variables. But since there are, as seen in the wikipedia page, the pattern I was looking for does not exist. Now it makes sense why Fermat was trying to prove there were none for exponent 2: that becomes the ‘odd’ case.
advances in solving the unsolved “Fermat” and other conjectures may be useful in real world problems.
eg The Fermat conjecture may be paralleling equations (involving squares…) that lead to prime numbers… Primes are useful for cryptography. and other problems…
Questions that have easy answers don’t get called “Fermat’s 953246th conjecture”.
It’s not that Fermat was trying to prove it; it’s that he (in a flash of insight?) had what he thought was a “truly marvelous proof.”
If mathematicians think you have a “truly marvelous proof” of anything, they’ll get all excited about it.
Mathematicians, especially theoretical mathematicians working in number theory, are not particularly interested in real-world problems. 
(Also, despite the importance of elliptic curves in cryptography, the math involved the proof of FLT, or more specifically the modularity conjecture, is light-years beyond anything involved in real-world problems, at least for the foreseeable future. It’s not unlikely that there will be some application eventually, but mathematicians aren’t interested in FLT because they’re salivating over crypto applications.)