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Main Answer I have a PhD in Mathematics and came across this question. To be honest, I dislike almost every single answer, except maybe L.Dutch's answer concerning Wile's proof of Fermat's last theorem. However, I do think there is a much, much better candidate, and one that would make every mathematician reading your story quite delighted: https://en....


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The Riemann hypothesis is what you're looking for. Basically everyone in number theory assumes it to be true (although no one can prove it). Variants of it have been proven in other settings. Many results, including entire theories of math, are conditional on its truth; these would all collapse if it was shown to be false. The discovery of even one ...


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Date of the next lunar/solar eclipse. See “Antikythera mechanism”: https://en.m.wikipedia.org/wiki/Antikythera_mechanism Eclipses follow such a complex pattern, they cannot be estimated from previous events, but as the Greek mechanism shows, they could be calculated.


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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 ...


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No such thing exists. All mathematics is a type of language. Like language, it looks mysterious to people who don't speak it. But if you study it enough, you will understand it. There are no exceptions. (*) Calculus was once an arcane branch of knowledge known only to Newton, Leibniz, and their handful of peers. It made them gods in terms of their ability ...


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This definitely wouldn't work in pure math, as there's nothing so "destructive" you could prove that wouldn't make you a celebrity among mathematicians. Finding a contradiction in ZFC would get you a Fields medal. Disproving the Riemann hypothesis would get you a Fields medal and a million dollars. There would never be any question about whether to publish, ...


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Instead of looking for new math that the children can do, show them learning math faster. All new math is built on the old. To do some incredibly complex proof, you'll typically need algebra, equations, maybe calculus or group theory or probability or what-have-you. The point is, it will be clear that these children are exceptional WELL before they invent ...


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Let's assume that this student wants to begin by understanding the twin pillars of modern physics: quantum mechanics and general relativity. There are several major tools in the toolkit of anyone studying both of these theories at a basic level: Calculus (single-variable and multivariable) Differentiation Integration Operators such as divergence, gradient, ...


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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 ...


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Estimate exponential growth using grains of rice and a chessboard. The story goes: The ruler or India was so pleased with one of his palace wise men, who had invented the game of chess, that he offered this wise man a reward of his own choosing and he said to the man: “Name your reward!” The man responded: “Oh emperor, my wishes are simple. I ...


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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.


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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?


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Where a cannon ball will land. A problem of extreme importance in the Middle Ages, why Newton and Galileo were studying gravity, and (because of the scale involved) nearly impossible to estimate with the precision desired by commanders. Because of the great mass of a cannon ball, the effect of wind and complicating factors of air resistance are ...


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A very well known conjecture (hopefully anybody with a CS degree would recall this, for example) is that there does not exist an efficient algorithm to take the discrete logarithm, in the most general case. This fact is sometimes used in cryptopgraphy. There are enough cases in which it is computationally tractable that it is plausible, at least, that your ...


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The following statement is unproven but widely assumed to be false: e+π is rational. A rational number is a number you can write as a fraction. For instance "0.25" is rational because it can be written as "1/4". "7" can be written as "7/1", "14/2" or whatever. We are absolutely sure that π and e are not rational, but mathematicians only assume that this ...


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For basic physics, you need (multivariable) calculus and linear algebra. This is basic literacy. You won't get anywhere in physics without them. There are some differential equations too, but one tends to learn that on a case-by-case basis as one studies examples. For general relativity, you need Riemannian geometry. Talking about curved spacetime only ...


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I find it hard to believe that a proof or disproof of any mathematical statement could cause "many careers in tatters". If professional mathematicians around the world all made the same mistake, it hardly reflects poorly on any individual mathematician. Moreover, when a mathematician makes a significant mistake, it almost always is for some non-trivial ...


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Take the mathematics needed to understand Wiles's demonstration of last Fermat's theorem. no three positive integers a, b, and c satisfy the equation $a^n + b^n = c^n$ for any integer value of n greater than 2. Without a master in mathematics you cannot even think of starting to learn the basis for it. The demonstration above is based on linking modular ...


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Decrypt an RSA-style Message Although the specifics of RSA encryption and decryption - or even more generally public key cryptography didn't come around until the 1900s, the ideas of factorization go back to the Greeks. Furthermore, RSA itself works on rather simple mathematics: multiplication and modulus operations. Wikipedia even has a simple example ...


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I, the King, wish to share the Kingdom's wealth with the People. If the Kingdom's population keeps growing, how long before they collectively are richer than the Royal Family? An estimate would say 'Probably 100 years'. An exact formula says never. Stick with me here. Let's say this is a verrrryy nice king. What goes around comes around- he shares his ...


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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) ...


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I think that without frame matching there will be severe limitations. While I am sitting at my desk to write this answer, I feel I am standing still. However: planet Earth is rotating around its axis, and this gives me a certain velocity vector $v_E$. This is 1668 km/h at the Equator, 0 km/h at the poles. the Earth is orbiting the Sun, this gives me ...


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Inter-universal Teichmüller theory is a real-life example of mathematics understood only by a handful of people, nearly all of whom are students of the guy who created it. There is a claimed proof of the abc conjecture which has so far been neither verified nor definitively disproven because the material is so impenetrable. https://en.wikipedia.org/wiki/...


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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 ...


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Peano Arithmetic (PA) is a very elementary theory of arithmetic. By Gödel's theorems, if it is consistent then it can't be proved to be consistent using methods which can be formalized in the theory itself. Nevertheless, it can be and has been proved to be consistent in stronger theories, so much so that the most mathematicians would regard the consistency ...


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Yes - you use a Base 6 system which has been done before Quite a few cultures used Base 6 counting systems in the past, and it is actually quite logical because it is the natural outcome of counting on one hand. As well as being popular a long time ago, it is even in use today with some native cultures, such as in Papua New Guinea, Congo and Ural Mountains....


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N-dimensional geometry, where n > 4. It’s very difficult for our regular human brains to cope with it, but may well have all kinds of useful implications for physics.


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The Archimedes eureka problem. Have a number of strangely shaped objects made out of different materials. The challenge is to work out which of them is made of the densest material. You can't estimate that as they are strangely shaped. You can't just weigh them as they are different volumes. The solution is to weigh them, then sink them in water and see ...


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The Sand Reckoner of Archimedes Archimedes of Syracuse was a Greek mathematician who lived in the 3rd century before the common era. He was probably the greatest mathematician of the antiquity Among many other things, he was interested in devising a notation for very large numbers. In order to present his suggestion for a system to represent very large ...


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There are a few different ways I would answer this question depending on how you actually plan to write this story. I will interpret it in a few different ways and give answers below. Is there a field of math learn-able by only a few individuals? No. Another answer already pointed this out, but the vast majority of human knowledge can be understood by ...


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