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Is there any conjecture or famous unsolved problem, that doesn't require much prerequisite knowledge and could be plausibly proven / solved by freshman?

My hero is average freshman in mathematics, that proves a famous conjecture by mistaking it for homework, like George Dantzig.

Maybe something in combinatorics if this reddit comment is true

A few years ago at my university the final test on combinatorics included some unsolved problems. The students were supposed to have enough insight to realize which problems were the easy solvable ones and which ones were a waste of time to try to solve.

One of the students (this course is usually taken by first or second year students) actually solved one of the unsolved problems. It wasn't one of the really famous unsolved problems included on the test, but his result was certainly unknown.

The conclusion was somewhat less dramatic, though: the professors thought about his solution for some time (months) and after they were convinced that there isn't a loophole in the argument, the result was published and eventually became the basis for this guy's PhD. research.

Problems in combinatorics are certainly easier to approach on this level, since their solution usually involves some clever trick, rather than extensive application of deep theorems.

ragac
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  • Do you want a solved problem (in which case Pythagoras’ Theorem will do you nicely) or an unsolved one (but no one can know how an unsolved problem can be solved; if they did, they could solve it). – Mike Scott Nov 03 '19 at 20:25
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    Unsolved, I will just write that it was solved / proven by my protagonist – ragac Nov 03 '19 at 20:26
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    Being a fan of Dirk Gently, the Moving Sofa Problem is one of my favorites. – Escaped dental patient. Nov 03 '19 at 20:35
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    List of unsolved problems in mathematics, dear old Wikipedia. – AlexP Nov 03 '19 at 20:37
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    This would make more sense on math.stackexchange.com – SurpriseDog Nov 03 '19 at 22:51
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    Would it be OK if it were a moderately famous only recently solved problem? (Not famous enough that non-mathematician readers would have heard of it, let alone hear the news that it was solved.) In that case, the sensitivity conjecture would be an excellent example. – Mees de Vries Nov 04 '19 at 06:31
  • There are lots of apocryphal stories like this. There is a true one about Dantzig but he was not a "freshman" at the time---he was a 1st year PhD student. – Kimball Nov 04 '19 at 09:13
  • @Kimball I know but the story works better with freshman. – ragac Nov 04 '19 at 10:01
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    For heaven's sake don't ask this at Math.SE. You will be downvoted into oblivion. Over the years we have seen to me "I wanna-be-famous-by-proving-this" posts containing nothing but non-sense. It gives people pimples. – Jyrki Lahtonen Nov 04 '19 at 11:06
  • If you take an unsolved problem, what impact will it have on your story if that problem is eventually solved? – kutschkem Nov 04 '19 at 13:23
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    I am a bit late but maybe this can be useful. Las year a 25-years old mathematical problem was solved by an anonymous user in 4chan. Probably he wasn't a freshman, but the maths involved were not that high – Ripstein Nov 04 '19 at 13:58
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    https://youtu.be/a1DUUnhk3uE?t=766 Famous puzzle... Unsolved apparently as the number of prisoners goes to infinity.... I would believe a freshman could solve it... I would believe he would think it was homework..... Probably not as impressive as you want it to be – kaine Nov 04 '19 at 15:18
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    You might also be interested in picking a very recently solved problem instead. For example, representing every whole number as the sum of 3 cubes. Several values are known proven impossible (n cannot equal 4 or 5 modulo 9), but there's an increasingly small list of unproved values (the only remaining unsolved cases up to 1,000 are 114, 390, 579, 627, 633, 732, 921 and 975; 33, 42, 906, 165, and 3 were all found just this year, most by the same team). https://en.wikipedia.org/wiki/Sums_of_three_cubes – Draco18s no longer trusts SE Nov 04 '19 at 16:47
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    Shortest superpermutations are another good one. Was only a year ago that a better upper bound on the length of the shortest superpermutation of n states was found. I'm pretty sure we have already found the shortest possible superpermutation for n = 4, and even if the sequence was already known in the setting of your story, having someone construct it from first principles would be impressive. – Draco18s no longer trusts SE Nov 04 '19 at 16:50
  • To be clear, by "increasingly small" @Draco18 still means infinite. :-) – Mees de Vries Nov 05 '19 at 07:18
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    When I was at university somebody was explaining the dual-marking system used on maths papers. To get a first-class pass you needed answers rated "alpha" in which case your actual mark was ignored. An apocryphal story was a maths paper with a misprint turning "routine" into "never proved", and a student who spent the entire exam trying to solve this one question and failing to do so. Undaunted, he spent the next two days in his room and then went to see the prof. he thought would be marking those papers. "Thought you'd like to see the rest of my answer". He got his first. – nigel222 Nov 05 '19 at 14:57
  • @MeesdeVries True enough, though my understanding is that both (a) larger values are generally easier to find solutions for and (b) we don't much care about looking for the solutions to larger values. ;P – Draco18s no longer trusts SE Nov 05 '19 at 15:40
  • Sounds like "Good Will Hunting". – computercarguy Nov 05 '19 at 19:53
  • @computercarguy The problem from "Good Will Hunting" is actually pretty trivial to solve if you know what the words mean, and the concepts could be pretty easily explained to anyone. I don't think that problem was intended to be an "unsolved" one, because it definitely isn't. It's only hard if you don't know the terminology. – Darrel Hoffman Nov 06 '19 at 19:05
  • @DarrelHoffman, I wasn't referring to specifically any one problem in the movie, just the movie in general. But yes, the problem on the hallway board was supposed to be an "impossible problem" for the level of education the students had. IIRC, the professor had to bring in other mathematicians to verify the solution Just because it wasn't completely unsolvable IRL doesn't ruin the movie or the idea. Nor does it prevent a parallel with this question. – computercarguy Nov 08 '19 at 17:24
  • @Draco18s: the Youtube Numberphile has a series of videos on the sum of cubes. 42 and some others are solved now (with the help of the audience and cloud-computing) – Taladris Nov 09 '19 at 01:18
  • @Taladris Yes, that's how I knew about it. – Draco18s no longer trusts SE Nov 09 '19 at 02:56
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    @Draco18s: great! I misread your comment and believed you said 42, 33, 3, etc were still unsolved cases – Taladris Nov 09 '19 at 04:05

9 Answers9

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Since you're interested in something currently unsolved, you don't really care about how protagonist will solve it, only that he does so, let's go for something extremely simple, simple enough that even a reader with only basics of mathematics can understand: Collatz conjecture (also known as 3n + 1 Conjecture)

"Consider the following operation on an arbitrary positive integer:

If the number is even, divide it by two.

If the number is odd, triple it and add one.

...

Now form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next.(Or in human language - keep repeating the previous process with new number again and again and again...)

...

The Collatz conjecture is: This process will eventually reach the number 1, regardless of which positive integer is chosen initially."

This is simple looking, yet currently unsolved problem, which looks simple enough for a reader who doesn't understand too much math as something solvable by a genius student.

A real student of mathematics or a professor will, reading your fiction, probably exclaim "That's not how it works, it's basically sure thing that any proof of this conjecture, if this conjecture is even provable, will be hundreds of pages long!"

But most of the audience will not find anything wrong with your protagonist solving this on ~6 pages as a homework. Only thing that might cause suspension of belief is that the guy has never heard of this conjecture before and confused it for homework.

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Failus Maximus
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    It's a pity the OP didn't ask about the solved problems. I was about to go seek the problems appropriate for each era, starting with the proof of the existence of irrational numbers for the ancient era, cubic equations solution for the 16th century, and then also mention modern https://mathsci.fandom.com/wiki/The_Haruhi_Problem solved by anon from 4chan – Failus Maximus Nov 03 '19 at 20:59
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    You don't even need a 6 page proof - you just need one example number that doesn't reach 1 because it goes into a loop. – Jerry Jeremiah Nov 03 '19 at 22:54
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    Colltz might be proved in just a couple pages if the proof links together a couple existing hefty proofs. Think about if Weil’s work unifying various domains had come before Taniyama Conjecture. Fermat might have been solved easily if Weil hadn’t realized the ramifications of his modularity work. – SRM Nov 03 '19 at 23:02
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    I'm liking this. And there's been enough study that we know it works for all everyday numbers. You counterexample could just be known as the "Collatz-counter" or whatever in-universe, and you'd probably never have to really qualify it. – Gloweye Nov 04 '19 at 08:33
  • @JerryJeremiah That would be debunking it, not proving it, right? – Mast Nov 04 '19 at 09:06
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    @Mast It would be proving it to be false, which is just as good. – Omegastick Nov 04 '19 at 14:48
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    @JerryJeremiah To nitpick: You would still have to give the number and show that it is indeed a proper counterexample. As far as I understand there are known lower bounds on the size of a circle, which depend on the size of numbers involved. And since anything small has been checked by computer, your counterexample would either involve a theoretical proof of its existence or listing a sequence of the hundred of thousands 20+ digit numbers which form a cycle, neither of which would come in below 6 pages. – mlk Nov 04 '19 at 15:43
  • For reference, the Collatz conjecture has been checked by computer for all starting values up to 87×2⁶⁰ and the lower bound on the size of the loop is 35,400 terms. – Draco18s no longer trusts SE Nov 05 '19 at 02:58
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    There's an ongoing volunteer computing project that is brute forcing Collatz using GPUs. They're up to 6.5 * 10^21 and have not found a counter-example yet. https://boinc.thesonntags.com/collatz/high_steppers.php – numbermaniac Nov 05 '19 at 08:06
  • ‘A real student of mathematics or a professor will […] exclaim "That's not how it works […]’ As a genuine professional mathematician (admittedly not in a relevant speciality), I wouldn’t even exclaim that. Finding such a proof for the Collatz conjecture or similarly elementary-but-very-hard problems would be exceptionally impressive — but it doesn’t seem a priori implausible, and you want to show the student as being exceptionally impressive, right? – Peter LeFanu Lumsdaine Nov 05 '19 at 12:46
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    I read that this problem was originally called the Hailstone Problem, because the value of the number rises and falls with each iteration, like a hailstone in a storm-cloud. Once it hits a power of 2, it plummets straight to the ground. – Oscar Bravo Nov 05 '19 at 15:54
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    @numbermaniac: a few of the answers on C++ code for testing the Collatz conjecture faster than hand-written assembly - why? suggest ideas for short-cutting the recurrence based on patterns of multiple low bits in a number. Hopefully the GPU "brute force" isn't totally naive, and is using provably-correct optimizations. At magnitudes like 10^21 I assume it must be. – Peter Cordes Nov 05 '19 at 17:40
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    @mlk Well, you just have to provide the 70-odd digit number that starts the loop and say "this loops after 7 million steps". Verification is going to be easy once you have the number. – Yakk Nov 05 '19 at 20:30
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Others have mentioned some famous conjectures such as the Collatz conjecture and P = NP, but I think it's awfully unlikely that a freshman math student would be able to solve such a problem. About the Collatz conjecture, Paul Erdős famously said that "Mathematics may not be ready for such problems"; and about P = NP, Scott Aaronson wrote that "any proof will need to overcome specific and staggering obstacles" and "we do have reason to think it will be extremely difficult."

Instead, I suggest a Diophantine equation. A Diophantine equation is simply any polynomial equation (that is, an equation built out of variables, constants, addition, subtraction, and multiplication), where the question is, "Can we make this equation true by setting each variable to a whole number?"

A simple example of a Diophantine equation is $x^2 + y^2 = 5$. This Diophantine equation has 8 solutions. One of them is $x = 2$ and $y = 1$. The other 7 solutions can be found by switching $x$ and $y$ around, and by negating one or both of them.

It certainly is plausible that a Diophantine equation could baffle mathematicians for years, but then be solved by a freshman math student. And I have an anecdote to prove it!

In 1969, D. J. Lewis wrote a paper about Diophantine equations, in which he wrote that the equation $x^3 + 5 = 117 y^3$ is known to have at most 18 solutions, but the exact number is not known. Two other mathematicians studied the equation and, in 1971, they published a short but difficult proof that there are no solutions. Finally, in 1973, another mathematician found an astonishingly simple proof of the same fact! The proof is:

The quantity $x^3 + 5$ is never a multiple of 9, but the quantity $117 y^3$ is always a multiple of 9, so there are no solutions.

(Source for the above two paragraphs: Gerry Myerson's answer on "Awfully sophisticated proof for simple facts.") Gerry points out that Lewis's equation, as printed, may have been a typo.)

So, although this particular equation was never famous, it did give some mathematicians quite a bit of difficulty, before, years later, someone found a simple proof that easily could have been found by a freshman math student.

So, which Diophantine equation should you use in your story? The paper "Some open problems about diophantine equations" contains one in particular which I think looks pretty promising. The problem that your hero solves could be:

Find all integer solutions to $x^4+x^2+y^4+y^2=z^4+z^2$.

Tanner Swett
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    And of course, the fictional proof in the story can't be so simple that the author would be expected just to exhibit it to the reader ;-) – Steve Jessop Nov 04 '19 at 23:17
  • @SteveJessop True, but you can do some 'camera' trickery. It was done in the film Good Will Hunting, see this QA, mostly by not-showing the whole process. The idea is to present a problem that is difficult to solve and show the character solving it quickly. – Draco18s no longer trusts SE Nov 05 '19 at 15:53
  • The problem discussed in that link from Good Will Hunting is not a great example IMO: the problem is fundamentally easy (accessible to strong high school students/early undergraduate, I would say) and just made to sound difficult by some unnecessarily complex jargon. – Mees de Vries Nov 05 '19 at 16:05
  • @MeesdeVries You aren't wrong that that particular problem is easy. But the story being told was about a 12 year old with little to no formal education. You want a problem for college graduate mathematicians, you pick something harder. My point was about how you show the characters solving it, rather than about the problem itself. – Draco18s no longer trusts SE Nov 05 '19 at 16:17
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    You may be correct about P = NP, but I would not be particularly surprised by some amateur finding a solution to the Collatz conjecture. Erdős only meant that we hadn't yet developed the right tool set to systematically investigate the conjecture. However, all that would be required to solve that particular problem is someone playing around and happening to notice a pattern that no one before had noticed. That isn't an unreasonable thing to happen, and is pretty much the same as happened with your $x^3 + 5 = 117y^3$ example. – Paul Sinclair Nov 05 '19 at 17:33
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    In case anyone else is also initially confused about the "x³+5 is never a multiple of 9" part, this helped me: If n is an arbitrary number divisible by 9, you can somewhat easily convince yourself by looking at it on paper that the results in the range of (n+1)³%9 to (n+8)³%9 are 0, 1, 8, 0, 1, 8, 0, 1, 8. After that it loops back, so you only ever have those remainders and plus 5 it's 5, 6, 13→4, 5, 6, 4, 5, 6, 4, …. – Fabian Röling Nov 05 '19 at 18:19
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    While the 117 thing is cute, it is also misleading. It was probably a typo (no proof of the "at most 18 solutions" was given). Someone took it seriously as a problem and didn't try a simple method (hence the complex proof). Then someone else said "wait a second". If the problem was "famous" you wouldn't have every mathematician making the blunder the first "proof" producer did. – Yakk Nov 05 '19 at 20:26
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    @Yakk That's true. Realistically, if a problem has already been tackled by hundreds of mathematicians with bulldozers and cannons, it's unlikely that a student will be able to solve it with a screwdriver and a BB gun. So I'm hoping that a less-than-famous unsolved problem will do the trick. – Tanner Swett Nov 06 '19 at 01:25
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Goldbach's conjecture states that every even integer greater than two can be written as a sum of two primes. If one could find a counterexample the problem would be solved (although currently all candidates smaller than the order of 10^18 have been tried). Alternatively if one could give a formula the problem could also be solved.

Dapianoman
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A perfect number is a positive integer N such that N is the sum of its divisors (other than itself). For example,

6 = 1 + 2 + 3

28 = 1 + 2 + 4 + 7 + 14

Question: Does there exist an odd perfect number?

Priska
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    Easy to answer in the affirmative just by finding an example. Hard to prove I’m the negative. – SRM Nov 03 '19 at 22:54
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    @SRM I wouldn't exactly call it "easy". – Priska Nov 03 '19 at 22:55
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    @SRM Can you find one? – Priska Nov 03 '19 at 23:06
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    No. But a student might plausibly luck into one. An easy proof mechanism exists as the basis for story. – SRM Nov 03 '19 at 23:14
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    @SRM I agree it's more plausible than many other problems (and one can suspend disbelief for stories). But it's not as simple as you make it sound: such conjectures have been thought about by many really smart people and tested numerically for numbers in a really big range. – Kimball Nov 04 '19 at 09:07
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    @SRM would be funny if the student just randomly wrote down a bunch of numbers because they couldn't figure out the answer and it turns out it actually turns out to be an example. – Erik Nov 04 '19 at 14:06
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    @SRM When you say "just finding an example", we know that any such example will have at least 300 digits (people have computationally checked all smaller numbers); so it's well within the bounds of a bignum package on a PC - but it's still going to be quite a bit of work to check. – Martin Bonner supports Monica Nov 04 '19 at 14:10
  • @MartinBonnersupportsMonica Agreed. And yet I've met people (mostly in math disciplines!) who idly pick at problems like these. – SRM Nov 04 '19 at 14:19
  • interesting. An odd number has only odd divisors, so an odd perfect number would have an odd number of divisors (so that the total sum is odd); it is possible for a number with an odd number of divisors for them to add up to more than the number (945 being the first example I found), so...this is a harder problem than it looks. :) – Skyler Nov 04 '19 at 16:16
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    @Skyler Wait, the only way a number can have an odd number of divisors is if it's a perfect square, because otherwise, for every factor X, there is another factor Y such that X×Y=your number, one must be above and one below the square root. If X=Y that means you have the square root of the number. So an odd perfect number would have to be a square. That limits your search parameters considerably... – Darrel Hoffman Nov 04 '19 at 18:18
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    @DarrelHoffman Not true, because 1 is counted as a factor, while the number itself is not. (Both of the [even] examples listed have odd numbers of factors.) – Skyler Nov 04 '19 at 18:34
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    @Skyler Oh, right, derp. I guess I just proved that an odd perfect number cannot be a perfect square, for whatever that's worth. Probably not much. – Darrel Hoffman Nov 04 '19 at 20:37
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    @MartinBonnersupportsMonica Minor note: the method uses for checking that there are no odd perfect numbers below a given bound isn't just doing a bruteforce computation but uses a lot of ideas which allow one to not have to check every example. 10^300 is really really big, and the current best bound is actually much higher. – JoshuaZ Nov 05 '19 at 01:19
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    @JoshuaZ D'oh! If I'd engaged my brain at all, I would have realized that brute-force checking 10^300 is not feasible. (If we could turn the whole of the observable universe into electron-mass computers, each of which could check a value in one pico-second, it would take approximately 10^180 years.) – Martin Bonner supports Monica Nov 05 '19 at 10:13
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You already have some good answers here, but let me suggest one other possibility that might work for you. Rather than an unsolved problem, you might look at a couple of cases in which there is a proof of something that can be stated simply, but the proof is unsatisfying in some way. Either the proof is so complex that it is accessible only to (extreme) specialists, or the proof is so vast that a computer is required to manage it.

Fermat's Last Theorem is in the first category and The Four Color Theorem is in the second. When I was a young mathematician, both of these were (widely believed) conjectures, but unproven. Now they have proofs.

But a simple proof of either, a proof whose details can be easily grasped by, say, a college student (and doesn't require a computer), would, itself, be a breakthrough.

Buffy
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  • The computer program that originally solved the Four Color Theorem by brute force was pretty simple. It's output was complex, but not its input. Would something like a clever program for solving some problem work for you, ragac? – SRM Nov 04 '19 at 14:16
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    Of course, the beauty of Fermat's Last Theorem is that it already WAS solved by an amateur - albeit an incredibly gifted one, but the proof was lost. Maybe your freshman could just rediscover Fermat's proof...? – Lefty Nov 04 '19 at 16:30
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    @Lefty, we don’t know whether Fermat proved it, only that he said he had proved it. Being gifted doesn’t make one immune to error (nor to falsehood). – WGroleau Nov 04 '19 at 17:44
  • @WGroleau True - but a little too difficult to explain in a comment. Nonetheless, my understanding is that he wasn't the sort of person to say such a thing lightly - and that he was right about everything else he discovered. – Lefty Nov 04 '19 at 19:58
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    @UlrichSchwarz Excellent! If only my accidental joke was deliberate, it would have been SOOOO clever. It must have been subconscious. Yes, that's what it was. Definitely. I can prove it, but there's not enough space left in th – Lefty Nov 04 '19 at 21:07
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    I wonder whether Fermat's proof ever existed. I suspect some time later he realized it was in error, or incomplete, and forgot to go back and erase his famous marginal note. Or if incomplete despite his best efforts, left it on purpose as bait for future generations of mathematicians. – nigel222 Nov 05 '19 at 14:50
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    @Lefty Fermat was good, but the suspicion is that he found one of a number of known "proofs" that actually have subtle flaws. – Martin Bonner supports Monica Nov 05 '19 at 18:02
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The biggest unsolved problem in computing is “can NP-Complete problems be solved in polynomial time?” NPC problems are a whole class of searching problems that we hit regularly in real world operations. Polynomial time basically means “a reasonable amount of time even on large problem sets”.

Most researchers think the answer is “no.” Proving “no” is really hard. Proving “yes” on the other hand just requires someone writing the program that does it. The student might not even realize they’ve done anything amazing.

If you do have your protagonist solve this problem, it’ll be a pretty serious kick in the pants for performance of all computing tech in your world.

Thorbjørn Ravn Andersen
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SRM
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    Note that other major algorithms have come from students and casual programmers just needing something to work. This isn’t without precedent. – SRM Nov 03 '19 at 22:51
  • Not to mention that it would make cryptography useless... – AlexP Nov 03 '19 at 23:12
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    @alexp Maybe. There’s good arguments why it might wouldn’t. See here for some: https://security.stackexchange.com/questions/12802/how-will-security-need-to-be-changed-if-p-np – SRM Nov 03 '19 at 23:21
  • The point is that it may, and most likely would. Suddenly all cryptography would have to be rethought. – AlexP Nov 03 '19 at 23:25
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    I agree it would cause panic and studying. The nightmare scenario would be a non-constructive proof. “I found that an algorithm must exist, but I don’t know what it is.” Then you’d have decades of everyone suspicious of everyone else and bluffing that “we have the decryptor!” Wouldn’t that be a frustrating world! – SRM Nov 03 '19 at 23:29
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    I see a few difficulties here. One, P = NP is so famous that he should’ve heard of it. Two, you can’t really explain the problem in a way that someone who’s never studied the theory of algorithms could get anywhere and write something that a computer scientist would agree is a rigorous proof of time complexity. Three, that’s notorious as a problem that crackpots think they’ve solved. Four, if the proof were in any way practical and not just of theoretical interest, the real-world and even philosophical consequences—for cryptography, AI, etc.—would be drastic. – Davislor Nov 04 '19 at 08:54
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    First: https://xkcd.com/664/ — Then, more seriously, https://www.scottaaronson.com/blog/?p=459, specifically the note “...that P vs. NP itself is a perfect example of a monstrously-hard problem for which we could nevertheless recognize a solution if we saw one—and hence, part of the explanation for why it’s so hard to prove P≠NP is that P≠NP…” – leftaroundabout Nov 04 '19 at 11:06
  • @SRM haha, a non-constructive proof for P=NP would be very delicate! I think this would actually do more to push CS and maths away from ZFC entirely, through being perceived as a paradox similar to Banach-Tarski (but much more severe). Related: https://mathoverflow.net/questions/50023/independence-of-p-np – leftaroundabout Nov 04 '19 at 11:10
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    @Davislor In response to points One, Two and Three: you could plausibly set a CS student a task to write a program that solves a NP-complete problem, and then when their solution is being graded, the professor looks at it, thinks "this is clever", looks at it some more and realises it's polynomial time. It's not quite the same as the request in the question, but it's pretty close. – Christopher Nov 04 '19 at 17:40
  • isn't this functionally the plot to Silicon Valley? – Erin B Nov 04 '19 at 18:38
  • @Christopher That’s a great idea! – Davislor Nov 04 '19 at 21:39
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To add on to the list of statements that are simple to understand: The Twin Primes Conjecture.

"Twin primes" are pairs of primes that differ by $2$, like $3$ and $5$, or $11$ and $13$. The open question is "Are there infinitely many pairs of twin primes?"

Dan Staley
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I would go with finding a non-trivial zero to Riemann's Zeta function, thus effectively proving (by counterexample) that Riemann's Hypothesis is false.

Pick a solution with the form $n/(2n+1)+xi$, with insanely large $n$ and very large but nice looking $x$.

Make it such that it is possible to simply check that the solution is indeed a zero of the Zeta function, thus the student would be easily believed, as false solutions to the problem pop-up very often, and even checking a correct proof could take years.

This problem fits two things that are nice for a narrative:

  1. It is said that the one to solve this problem is destined to mathematical greatness.
  2. Even a wrong or fuzzy method could still provide a verifiable non-trivial solution, and thus it wouldn't take long before the consequences of the discover start taking place.

The downsides are:

  1. It is a rather complex conjecture to be understood by someone who only knows about high-school math. (probably most of the potential audience)

  2. The hypothesis is widely believed to be true. (though it being proven false would have a much bigger impact).

Another "solution" would be finding a polynomial algorithm to find hash collisions for SHA2, as far as I know, there isn't a proof that such an algorithm would be impossible to exist, but a lot of cryptography based systems rely on the fact that it is too computationally expensive to create arbitrary hash collisions. This may or may not create different ramifications for the story, but it is something that a teacher wouldn't ask in a test as a "unsolved problem", it might be part of some research task to find methods that are better than brute-force (slightly, as has been done with other hashing algorithms), then by accident finding something much better than brute-force, or some algorithm that "just happens" to run fast very often, even if it could take very long (akin to Quicksort algorithm).

Mefitico
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I would argue: any problem that can be conveyed simply without excessive higher-level math would be fair game.

Keep in mind, solving a conjecture/problem often doesn't depend on extreme math proficiency, but being able to creatively approach the problem from an unusual angle that hasn't been attempted before.

Here's a great example, from the 2011 International Math Olympiad. It was given to the brightest students in the world, yet this problem stumped most of them. And it wasn't because the problem required extreme proficiency with mathematics, but reaching an understanding of a core dynamic of how the system worked - and then simply exploiting part of that dynamic. The actual 'math' of the situation is almost an afterthought.

So let me give you some hypothetical examples:

"Yeah, I solved the 3 body physics problem. I mean, I know we were supposed to integrate over position, but that wasn't working out for me. So I tried to integrate over time and distance squared and I figured out that a lot of the complexity just disappears."

...

"Yeah, I solved the Collatz Conjecture. I thought to myself: why use a solid base? I mean, instead of each digit representing a constant number to an increasing power, why not make the digits correspond to prime numbers? So the number '3011' really represents 3x5+1x2+1x1. And when you look at the patterns in that number scheme? The patterns are pretty obvious - the conjecture is pretty trivially obvious."

...

"Oh, RSA decryption? Yeah, the whole 'Cannot find large factors' seemed kinda weird. I mean, we can express the large number in whatever base we want, right? So why not convert that large number into each of the first dozen or so prime bases. Then we guess at the lowest significant digits of the solution in each of those bases, correlate all those prime bases' guesses together, and get a pretty accurate picture for what the divisor must be."

Note: None of those hypotheticals are actually true. But it shows how an oblique, non-standard approach to a problem might be what actually 'cracks' the conjecture.

Kevin
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