A simple proof of the Gaussian correlation
conjecture extended to multivariate gamma
distributions by Thomas Royen: https://arxiv.org/pdf/1408.1028.pdf
1. Gather all the hay in multiple trash bags (this assumes that the needle remains with the hay and doesn't fall out)
2. After gather the hay, use a roller-magnet pickup tool over the area the hay was at to find the needle if it fell out of the hay (assumes needle is made of ferrous magnetic material)
3. Place each bag, one by one, inside the torus of a CT scanner. Turn on CT scanner.
4. Remove bag of hay. Check inside of CT scanner torus for needle.
5. Repeat as needed with other bags.
Alternatively, one could set the hay pile on fire, then run over the ashes with the magnetic pickup tool.
Case in point: Fermat's Last Theorem, or Collatz Conjecture
"Trivial" = "regularly covered in undergraduate courses"
"Easy" = "a good topic for an undergraduate honour's thesis"
"Non-trivial" = "a good PhD thesis topic"
"Distinctly non-trivial" = "a groundbreaking result which will establish a professor's reputation in the field"
This proof is a perfect example of that statement. Formulating the problem the right way -- which is most often the hardest part -- was the real challenge here, not the mechanisms needed to do the formulation or the proof.
Let y(x) = a*x^4 + b*x^3 + c*x^2 + d*x + e,
whereby e is not _necessarily_ the base of
natural logarithm> mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book."
I had an undergraduate course my freshman year where we went through a circular proof of the equivalence of twelve or thirteen formulations of the axiom of choice. A hundred years ago, proving many of the steps of that proof might well have been non-trivial, perhaps even distinctly so.
It is simple to say E=MC^2.
That doesn't take away from the fact that it was very hard for it to be discovered.
And it's a good thing. It's a good thing that things which were hard to discover can be communicated easily. Otherwise we wouldn't be able to compress thousands of lifetimes worth of discoveries into a one-semester lecture.
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(+) footnote: well, unless P=NP.