The Erdos Paradox

For example: https://www.quantamagazine.org/amateur-mathematician-finds-s...

No way you'd get research money for this, but an amateur can sit and poke away at it.

These are all unsolved: https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_m..., and plenty look like they may be solvable without advanced math, just someone dedicated and a unique approach.

Particularly the Collatz Conjecture, afaik basically zero progress has been made on it, no "advanced" math topics are known to apply at all. So perhaps a unique angle is all you need. (Or maybe it's false, in which case all you need is a counterexample. Sometimes it happens, see e.g. https://www.ams.org/journals/mcom/1988-51-184/S0025-5718-198...)

Here's one example. In Darwin's "Origin of Species", a key principle which Darwin talks about at great length is the so-called Knight-Darwin Law, the observation that even in apparently asexual species, sexual reproduction still occasionally occurs, if only after long spans of time. In an article in Acta Biotheoretica, I argued this is in fact an infinite graph-theoretical principle. If G is the directed graph of all organisms, where an edge from x to y means that x is a parent of y, then (Darwin theorizes that) there is no infinite directed path through G all of whose vertices have indegree 1. (A vertex has indegree 1 if it has exactly 1 parent.) The low-hanging fruit (some of which I picked in my paper): Investigate the mathematical implications of the above property (esp. if G also has other biologically motivated properties).

The Knight-Darwin law seems to have vanished completely from the literature around the start of the 20th century, until I cited it. Almost as if no-one has actually been reading Darwin in all that time.

To generalize: If you want to find low-hanging fruit, read non-fiction classics like Darwin and constantly be asking, "Is this some unrecognized mathematics in disguise?"

In his 2017 Christmas lecture (https://news.ycombinator.com/item?id=15898919), Donald Knuth mentions at around the 5-minute mark:

> You know, people think that mathematicians have been working for hundreds of years, and now there's tens of thousands or hundreds of thousands of mathematicians all spending every day working on problems, so how could you possibly still find a simple down-to-earth problem that hasn't already been studied, you know, way too much? And the answer is: those problems aren't rare at all, they keep coming up several times a year, and this is an example where even this very basic problem of rectangle into rectangles has all kinds of stones not yet turned.

You can see in the talk where he starts simple and just keeps working on it, until he gets something that seems new (a conjecture, which he proposed as a problem in the American Mathematical Monthly). Of course, this particular problem is now solved, but if you just keep exploring you'll pretty soon hit the limits of what's known. (Potential exercise that I haven't tried: load random sequences on OEIS http://oeis.org/ until you hit one where there are only conjectures, and not much research.)

Well, I can't tell you low-hanging fruit that's still out there, because if I knew it, I'd be working on it instead of telling you. :) But I can tell you some low-hanging fruit that I've done!

I'll stick to just one example, this[0] paper of mine was, like... I was astonished this was not already in the literature -- everything in this paper is easy to prove once you think to ask the question -- but as best as I and anyone else could tell, somehow nobody had thought to ask the question despite having asked other extremely similar ones. It's especially astonishing that Jacobsthal's exponentiation was independently rediscovered multiple times, but what I call "super-Jacobsthal exponentiation" seems to have only been considered once before and only briefly as a tool for something else (nobody thought to write down its algebraic laws). So, I wrote it up to make sure it was out there.

(Honestly I'd say a lot of my work has been LHF, I got through grad school while learning hardly any of the heavy machinery one normally does, unfortunately most of it isn't written up yet and so I can't really easily point to it.)

[0]https://arxiv.org/abs/1501.05747 (This, um, will require some knowledge of ordinals to read.)

I think it's not just a matter of low-hanging fruit, but low-hanging fruit that's interesting to other people and applicable (to other research or to the real world)

My favorite example of (formerly) low-hanging fruit is the formula for determining how often you can fold a piece of paper in half. Is there anyone who didn't grow up "learning" that you can't fold a piece of paper in half more than six or seven times? Until high school student Britney Gallivan found the formula, with basic algebra, and extended the record to twelve folds. [0]

[0] https://en.wikipedia.org/wiki/Britney_Gallivan

Another fun one (from the many examples above!) is a new paper I am working on at the moment as well, but in the intersection of physics and math: it turns out you can phrase a bunch of physics problems as optimization problems, then use some very simple tools in optimization to derive fundamental limits of how well any device could ever perform.

The preprint is here: https://arxiv.org/abs/1811.12936

An associated blog post is here: https://guille.site/physics-impossibility-results.html

Perhaps people who think mathematics is full of low hanging fruit are the same ones who grew up looking for binomial coefficient identities and calculating the 3n+1 sequence. It might not seem low hanging to everyone, but maybe to mathematicians.

See my reply to cjohnson318 for an example.

Very recently the superpermutation problem [a.k.a "haruhi" problem]. Obscure enough that a anime-loving anon gave a proof for improved bound that went unnoticed for some time.

Sure. But this was not the case with Szemeredi's work. It was recognised as great immediately, and Szemeredi was below 40 at the time. The problem was that his combinatorial approach was not considered to be deep enough to merit the medal.