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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....


77

About 30km/s; that it to say, the velocity cannot change at all if you wish to maintain the same orbit. Every circular orbit is associated with exactly one orbital velocity. Every general elliptical orbit is associated with exactly one velocity profile--one specific apoapsis velocity, one specific periapsis velocity, and one specific curve in between. If ...


77

Dogs. https://www.navysealmuseum.org/raven-nsm-multipurpose-canine If the problem is the limitation of human senses, dogs are the answer. Dogs have gone to war since there were dogs, and commandos and dogs work well together. Illusions designed to trick humans will be obvious fakes to the dogs. They will smell right through disguises of any sort. The ...


72

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 ...


69

Nothing special about 10. It's likely that the aliens would develop a system with a radix (base) of whatever number of fingers/toes/tentacles that they had easily available to count with. And as TheBlackCat pointed out, there are many options beyond that. There's even different base systems among humans. For instance, the Babylonians used a base-60 system, ...


61

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.


59

You imply that mathematics is some fundamental root of the entire universe. While there are many who agree, this is not a fully agreed upon assumption. Many would say what you describe just any other world, only with different math, that looks feels and tastes just like this one, because mathematics is nothing more than a human construct. (as SJuan76 says ...


52

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 ...


52

As far as you want. Humans have trouble conceptualising large numbers. We can’t count them, so above a certain point we see the number ‘178654’ and our brains turn it into ‘many’. Doesn’t change the value of the number, just how we intuit about it. For anything bigger than the number we can imagine (varies from person to person) we start doing maths rather ...


47

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 ...


46

I think the question is not so much what we can do without zero, but how zero could remain undiscovered when humans begin to advance. One of my favourite quotes from one of my favourite contemporary mathematicians (Roger Penrose) is that it's always possible to create an equations from numbers of a given type whose answer falls beyond that type: Positive ...


46

The biggest visual clue that you're on a much larger planet than Earth is a mismatch between atmospheric haze (related to distance), perspective, and horizon distance. A ship out at sea, for instance, would have much more perspective "shrinking" and haze coverage while still above the horizon than you're used to. If you have any ability to measure distance,...


44

The big clue is that you'll never see something disappear over the horizon. On an Earth-sized planet, the horizon is about 5 km away. Under ideal atmospheric conditions, you can see things up to 300 km away. Seeing something drop below the horizon is no problem. On a Sun-sized planet, the horizon is now 550 km away, but atmospheric conditions are no ...


43

It strongly depends which area of maths you're talking about. Category theory is basically new, so before the 1950s or so, it just didn't exist in anything like its modern form. Combinatorics has been around for a long time, but before Erdös it looked very different. Before Newton and Leibniz, the notion of calculus wasn't very clear, and its notation would ...


39

Absolutely not. Even on Earth, we routinely use other bases. Computer scientists use binary (base 2), hexadecimal (base 16), and octal (base 8), as well as decimal, very routinely. Various world cultures (past and present) have used number systems with all kinds of bases. There is a base-12 number system called Dozenal (or Duodecimal) that has some real ...


38

Technically, if he went all the way back to the beginning (where it was just starting), he won't have to relearn anything (as nothing had been invented). He can simply use his own knowledge, and it will become the norm - others will learn whatever he writes, and he won't have to learn anything new.


37

Probably not While it is true that a Base 8 system would mean that the general public would be able to work with programming numbers and hexadecimal better, anyone who uses those number systems with any frequency very quickly acclimates to the various base systems within any programming language. There are programmers who can just offhand convert numbers to ...


34

I'm interpreting this question as "Is there a large(ish) natural number, the knowledge of which is evidence of advanced maths, and which is in some sense universal" The theory of groups is of fundamental importance. It arises naturally from the analysis of symmetry, a basic property of nature. Among the groups, some have no normal subgroups, and such a ...


34

Your password system won't work on a flanking enemy. They will kill you why you are asking the password. IR visors will tell you where there are objects warmer or colder than the background. Those happen to be humans. Make sure that your team wears a clearly identifiable mark visible in the IR, so that if they see sergeant John Doe without the mark they ...


32

I think the knowledge would simply be lost in time unless they actually have use for it. A tribe of post-neolithic settlers have very little use for differential equations and stuff of that level. And even simpler stuff like linear algebra would not be helpful to them. Basic math is easily applicable in everyday life. Even in simple non-currency-based ...


31

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, ...


30

Something like a Vigenère cipher with a very long keyword. If the keyword is long enough, it approaches a "reused one-time pad". While reusing an one-time pad is a bad thing, proper exploits require a lot of computer power that might not be available to the attacker.


30

If the Sun were more massive than it is now, then the Earth would have to move faster to maintain the orbit that it currently has. The formula for the rotational velocity is $v^2$ = (G • M) / R, where v the velocity, G is the gravitational constant, M is the mass of the sun, and R is the radius of the orbit. As an example, if the Sun had four times the ...


29

Tic-Tac-Toe, or any other solved game that lets either player force a tie. Solved games, especially easy ones to remember like tic-tac-toe, can be pretty boring because both players can play their optimal winning strategy regardless of what the opponent does. This means if both players of a game of tic-tac-toe are playing optimally, then it will only ever ...


28

The use of prime numbers in communication is talked about in Prime Numbers and the Search for Extraterrestrial Intelligence. Here's one method: Create a rectangle. Divide it into units, such that each side has the length of a certain prime number. Encode images into the rectangular grid by making each square black or white (or a dot or a dash). Take apart ...


28

Volume is the cube of length. Therefore, if the length of a human drops by a factor of x; the volume drops by a factor of x$^3$. Therefore, a 3" person is $$\frac{3}{72} = 0.042$$ the length of a regular person, then he would be $0.042^3 = 0.000072$ times the volume (and mass) of that person. Multiply that factor by 200 lbs to get 0.014 lbs; or 0.2 ...


28

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 ...


27

Step back from the context of worldbuilding for a moment, and look at this from the context of storytelling. What purpose does the number system serve in your story? In-universe a different base might have advantages, but does the reader care about that? Are you trying to extol the virtues of a dozenal system to your readers? Is the difference in number ...


27

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, ...


27

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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