Every Unsolved Math problem that sounds Easy 

I have proven the Riemann Hypothesis but I'm currently busy feeding my cat.

I remember in my intro to proofs class seeing Goldbach’s conjecture on a problem set and being really frustrated I couldn’t figure out a super simple looking problem. I looked it up just to find I’d spent over an hour on an open problem lol

riemann hypothesis definitely does not sound easy.

yeah, apparently the birch conjecture was so easy and trivial that the statement of it is left as an exercise to the viewer

When Euler says he cannot prove something, the math community shivers.

My favorite unsolved problem in mathematics is the Moving Sofa Problem. Say you need to move a couch (or any 2d shape) around a corner in an L-shaped "hallway" of unit width. What is the maximum area of the couch and what is its shape? This problem was proposed in the 60s and we have a very good approximation, but no exact solution yet, at least not one that has been proven.

<a href="#" class="seekto" data-time="332">5:32</a> btw, it's been proven that Yitang's method will not work to bring it down to 2. 6 is the proven minimum of the method.

<a href="#" class="seekto" data-time="510">8:30</a> You literally forgot to say what is Birch (...) conjecture You just said it has something to do with elliptic curves

For the kissing numbers, I assume the reason 8 and 24 is because Maryna Viazovska (and others) solved the sphere-packing problem in those dimensions using the E8 lattice and the Leech lattice back in 2016 - it made the news. A well-deserved Fields medallist.

2 small mistakes that I noticed: 1 - recent research has shown that there are infinetely many primes with distance at most 2 houndred and something (cant remember exactly). But they havent proved for 6 yet. We can only lower the constant to 6 if the Riemman Hypothesis is true. And according to Terence Tao, we're not close to improving this result. Any further result would need a huge math breakthrough 2 - It's not really a mistake, but something important that you missed. You don't need to go as far as algebraic numbers when the matter is pi+e. We don't even know if they are rational, and neither pi.e or any pretty expression that you could make with them

Leon-hard? Leon... HARD? Is that how he said his name? No really, this is more important than the unsolved mysteries of math

Collatz conjecture is basically a race. It’s all about proving you always hit a power of 2, and while there’s infinitely many of them, they also get spaced out up the number line while 3n+1 tries to keep up.

Every time again and again I see a picture of Hermann Grasmann when people talk about Goldbach. Goldbach wrote to Euler, yet we have a photograph of Goldbach and not one of Euler. That's because the first photocamera wasn't invented for like 60 years after Goldbach and Euler died. We do not know of any depiction of Goldbach, and people just take the first image that pops up when you google Goldbach and are like "Yup, that's him. No more research needed." It's so sad to see it happen every time :(

"Math problems that sound easy" Problem #1: So... There's these 24 dimensional Spheres...

whats crazy is that theres a chance that some of these theorems are completely unprovable but are, regardless, true. so yes, euler was a (leon)hard mf

Famous conjectures which are easy to understand: There are infinitely many perfect numbers All perfect numbers are even

You kind of brushed past one of the things that is so valuable in the Riemann zeta hypothesis. Solving it gives us some more structure in the primes. It has been shown already that zeta(s) is also equivalent to the infinite product, over all primes p, of ( 1/(1 - p^(-s)) ). Edit: noting that by setting this equal to the infinite sum fornula for zeta given in the video, we get a correspondence between the natural numbers and the primes.) Solving Riemann could give real insight into the distribution of the primes. As I tried to point out above, there is a very direct link between the Riemann zeta function and the distribution of primes. Cool video. Enjoyed it a lot.

The riemann hypothesis and *especially* the birch swinnerton dyer conjecture are completely impossible to understand for 99% of people. They definitely are not simple looking :|

Definition 1.0: We define the phrase "sounds easy" to mean "doesn't sound easy".

<a href="#" class="seekto" data-time="518">8:38</a> "it's the only other one we can remotely describe in plain English" P vs NP can be easily described: P is the set of decision problems that can be solved in polynomial time and NP is the set of decision problems whose solutions can be verified to be correct in polynomial time; the problem asks whether those two sets are identical. The only further issue is how to define "decision problem" (easy, it's a problem with yes/no answer) and how to define "polynomial time" (a bit harder: a solution that runs in polynomial time has a number of steps proportional to a polynomial P(s) where s is the size of the input in bytes; here "input" is, for example, for the unknotting problem described previously, a description of the knot you want to decide whether it's identical to the unknot. We can easily verify that a given way of moving the folds proves that a knot is the unknot, so this problem is in NP)

<a href="#" class="seekto" data-time="415">6:55</a> Matrix multiplication is not n^3 complexity. The naive implementation is, but the best currently known is about n^2.37.

No unsolved math problem sound easy to me considering the geniuses that already tackled them 😅

A lot of these problems seem to revolve around finding a way to relate sums to primes. I wouldn't be surprised if one day someone found a way to relate them and quickly solved many of these problems using it.

The Lonely Runner Conjecture sounds as if it can incorporate the aspects of music theory in order to solve it. That is, each runner can be given a set BPM. Another idea could be equally separated notes in terms of rhythm.

obviously gamma is rational, It's just gamma/1

great stuff, thank u for posting! <a href="#" class="seekto" data-time="105">1:45</a> leon-hard lol

Two things that make these unproved math conjectures so captivating. 1) Godel's incompleteness theorem suggests that there exist true conjectures that cannot be proven. ('suggests', there may be alternative systems that can prove a conjecture that is unprovable in a different system). 2) The Poincaré conjecture was one of the millennium prize problems that did get proven, which does demonstrate that not all of these conjectures are unprovable. We'll really never know if these problems are unprovable. We only know that they are not unprovable once they are proven.

Goldbach conjecture: 2n = p + q could be restated as n = (p + q) / 2, n >= 2, p and q primes: every integer greater than 1 is the average of two primes.

Regarding lowering the upper bound on prime gaps: Assuming a major unsolved conjecture to be true is not a "subtle technical assumption". A technical assumption in mathematics is a restriction of the general problem to a subset of special cases (which can range from covering only a few finitely many cases up to "almost all cases") for which the problem can be explicitly proven. Their unconditional bounds are an astounding achievement. Going from 246 down to 6 on the other hand is "nice" and may point to new ideas, but may just as well turn out to be completely hopeless. BIIIIG IF.

Bro you cant even state the Birch and Swinnerton-Dyer conjecture without technical definitions so i don't know if it sounds easy to anyone. For the curious, it says that the rank of an elliptic curve of a number field is the order of vanishing of its L function. For this to even make sense, you need to knoe the Mordell-Weil theorem (not too hard) and the Modularity Theorem which Wiles proved to conclude Fermat's last theorem. BSD is very deep

leonHARD Euler

If you think we wouldn't understand the Birch_Swinnerton-Dyer conjecture, fine, but just leave it out. No point in starting an explanation, and saying that elliptic curves have certain properties, but stopping there as if you've taught us anything. We're none the wiser.

It seems like trying to prove the Colatz conjecture will only get us stuck with circular arguments

Thank for saying, what the Birch and Swinnerton Dyer Conjecture is. And so easily to understand too.

I have been a subscriber for three days and I really like your content ;)

A lot of the problems seem to revolve around the wierdness of the integers. For the real numbers we have things like calculus which look at functions over the real numbers and things like limits and derivatives. But for the integers we have a field studying them called number theory however many unsolved problems come from there (including goldbach and collatz) as we don't have much knowledge on that field and don't understand things like the prime factorisation of n vs n+1 or the distribution of primes.

This dropped exactly when I needed it. Subscribed immediately. Excellent work.

One interesting thing about Goldbach's conjecture is that a slightly weaker form of it has been proven. That is, every odd number bigger than 5 can be written as the sum of 3 (not necessarily distinct) primes. This follows easily from the normal ("strong") version of the conjecture, because every odd number is just an even number + 2. So if you can write every even as p + q, you can write every odd as p + q + 2. Going from the weak form to the strong form is difficult, though, and an unsolved problem.

In the table at <a href="#" class="seekto" data-time="52">0:52</a> for the kissing number problem, the correct numbers are known for dimensions 1, 2, 3, 4, 8, and 24; also, the numbers for dimensions 5, 6, and 7 are believed to be correct, though not proven. Noting that the numbers grow exponentially and never more than double, isn't it safe to say the lower bound for dimension 23 can be increased to 98,280? I get this number by taking half of the number for dimension 24, which is known to be correct.

very great video Idea! I've been searching for such a list!

The table at <a href="#" class="seekto" data-time="420">7:00</a> is confusing. The decision versions of the problems on the right are NP-complete, in particular this means that it is an open problem (called NP=P?) whether they can be solved in polynomial time. Also technically in Computational Complexity Theory, problems are "Exponential Time" if they can be solved quickly enough, in time at most exp(p(n)) for some polynomial p, where n is the size of the input. A problem that cannot be solved in polynomial time is superpolynomial, and a problem that cannot be solved in exponential time is superexponential.

The Goldbach Conjecture solution - with some assumptions, 3 probably being the deal breaker: 1. Any number (and by extention, any even number) can be described as an addition of 2 numbers, with either: both of them being exactly half of the sum; or one of them being smaller than half the sum, and the other being larger than half the sum (this includes one being equal to the sum and the other being zero, and one of them being larger than the sum and the other being negative) 2. There are infinitely many prime numbers. 3. There does not exist a "gap" in prime numbers, where there is no primes between half the sum, and the sum. 4. Two odd numbers always add up to an even number. 5. All primes larger than 2 are odd.

The P vs NP problem doesn't sound too difficult either: give an example of a task that can't be computed quickly but if someone gives you a solution, it can be verified quickly (quickly == the number of steps is a polynomial of the input size).

Great vid! Love how you broke it down.

you can't take a math problem like the birch and swinnerton-dyer conjecture, decide it's too hard to state, then put it in a video like this anyway lmaoo?????? god that blindsided me, gave me a good laugh thanks dude there's actually quite a number of decent unsolved problems that would be better to mention, such as whether it's possible to make a 3x3 magic square out of 9 distinct square numbers. not nearly as famous as anything on this list

Honlestly, you only need the names Erdos and Euler and you get a truckload of easy looking open problems.

Honestly great video and explanation of the various problems. Many I had not heard of before. You did it a little bit with the Collatz conjecture and the Kissing number but it would have been awesome to get a little bit more insight into why these are actually also interesting problems or what benefit solving them could have for the real world.

Bravo! for putting this together...

Great video! Topics were explained really well!

this is the most american way of pronouncing euler-mascheroni lol

damn that is gamma a rational number hit hard, recently learned about it in my discrete maths class and would never suspect it to be such a mystery, great video

umm did you juat skip over swinsgrns dyer conjecture? not explaining it?

Great video - I may have been you 5,000th subscriber as when I hit subscribe it went from 4.99K to 5K!

As much interesting this quest of search for answers is for our curiosity, we should stop asking for "is this and that linear combination of transcendental numbers an algebraic number or not?" because already there are countably infinte many ways to linearly combine Pi and e, so as soon we add even more Irrationals, there are even more linear combiations of them, all are infinitly many. So this question can go forever if we not stop asking if these and that linear combinations of Transcendentals yield rational or irrational numbers. Side note: In one of your in-video-pictures, you show <a href="#" class="seekto" data-time="469">7:49</a> and thanks for the fact that 3,3030 0300 0300 0030 0000 3000 0003... is non-repeating and therefore irrational, I had not realized until now that EVEN if we know all of the infinite digits of a number, it can still be without repeating digit-sequences. Now I know that we can just as easily (as we create or think of resp. Rationals) create Irrationals with also knowing all digits by just defining a non-repeating and infinite digit sequence like "each 3 you pass, put one more 0 between it and the next 3 with increasing string-length of just FINITELY repeating 0-s" and its non-repeating, analogously with any other strictly monotonous, whole-number-for-y-yield function of x="number of 3-s before the given digit in total" and y="number of 0-s until the next 3" and then expanding by exchanging 0 and 3 with one of the 44 other pairs of nonequal decimal digits or puting finite amount of other digits inside the above-created irrationals. (The monotony part is not needed if we still can be certain that no y value does ever repeat i.e. for a injective function, but monotony can more easily be understood, most easy for f(x)=x) Or just doing sums again, the all-infinite-digits-known irrationals should be more easy to decide if their linear combinations are rationals or not. For example, adding 4,4404 0040 0040 0004 0000 0400 0000 4... to the above irrational with only 3s and 0s, we get 7,7434 0340 0340 0034 0000 3400 0003 4000 0003 4000 0000 3400 0000 0034 0000 0000 0340 0000 0000 034... that is again irrational, as it does thesame with 34 as was with 3. Even though we could use any injective-whole-value-function for the amount of 0-s, using monotonous functions makes it more obvious to aknowledge this function by looking at the number, most easily with f(x)=x as I used above. Even more easily, we take only the injective function and instead of transforming it into amount of 0-s, just write all consecutive values as their finte digit-sequences and than back to back as decimal digits of a single number. This leads to, for f(x)=x+1, to my new favorite number. It is 1,23456789101112131415161718192021222324252627282930313233343536373839... as it lists all digit-sequences of natural numbers as its own digits, but is itself all-digits-known-irrational. But I again not know if its also transcendtal or algebraic. PS: You said there are more Irrationals than Rationals, is there a mathematical proof or could the Cardinality of both sets be the same?

<a href="#" class="seekto" data-time="564">9:24</a> man i must get that one million reward, I’ve always loved that fragrance!!

For all those asking about the Birch & Swinnerton-Dyer conjecture, there is an excellent book that explains what it is, assuming you only know high school maths. It is Elliptic Tales, by Avner Ash and Robert Gross.

I have found a proof for the Riemann hypothesis, however discovering my proof is left as an exercise to the reader.

The thing about irrational and trancendental numbers is, their definition is "complementary" or rather, x is irrational if it's *not* rational We don't have methods for identifying irrational or trancendental numbers, so each number has to be proven not rational or not algebraic

Watching this video makes me happy

The reimann-zeta function is only defined for that when re(s) > 1. Outside of that, it's just the magical "analytical continuation" that might be a bit much for this video...

For 3x+1 you can simply understand it tends too sink lower and higher spurts of going up are more common as you got higher in numbers but go back down too 4 2 1 If you th8nk ab9ut how our counting system is infinite you realize thats the going upwards in numbers will be more common And going back down will be less common eventually probably wnding at omega which is infinity+ 1 And in order too get a pair too do so would have too reach at a minimum of (W+1)/2 So in order too solve 3x+1 you would just have too solve omega Using other counting systems like binary the equivalent tok omega would be an infinite number of 1 in digits adding 1 would eventually make it equal 0 through basic computation and can be prooven that W = 0 Which through this would state That negative numbers are past(W+1)/2 in our counting system making number lines really just being a portion of a circle and probably making our graphs a 4d sphere How ever the point still stands If there us ever another looping chain in the 3x+1 problem it would be literally impossible too find due too it would have too cross the (W+1)/2 border both ways and have too loop Doesnt mean it doesnt exist but i believe there is a strong arguement too the fact that there is not a nother chain in the if odd you do 3x+1 and if even you divide x by 2 Than 4 2 1 aswell as the 3 other prooven chains due too it gravitating towards 0 and W which is 0 from the context of negative numbers in all retrospect.

I solved most of these back in high school

don't you dare to stop. you are doing great

I thought Tao basically showed there is No unbounded sequence of Colletz numbers, but that doesnt eliminate the chance that there is a loop

What’s the most efficient way to pack unit spheres in 4d space? That’s also unsolved, surprisingly.

Collatz conjecture could be restated as: for every integer, every Collatz sequence reaches a power of 4.

Hello Everyone, As a part of my Masters Programme me and my friend had written down a proof of Goldbach's conjecture using Graph Theory , the panel that we presented it too did not challenge it, but I'd really welcome and appreciate someone looking through.

that's the wonkiest pronunciation of "leonhard" i've heard yet

Well-made. No intro makes it feel like long-form tiktok haha

Related to kissing numbers (very vaguely): what is the smallest square that can contain N non-overlapping squares of unit size? We don't know what is the minimal size length for N=11,12,13 and most higher N.

I feel like the use of comic sans in this video was a deliberate and calculated act of evil.

I like that some of these math problems are covered by Varitasium. If you want a deeper explenation you should definitely visit his channel.

Random personal fact: Swinnerton-Dyer is my mathematical grandfather on the Mathematical Genealogy Project.

Another famous one: We don't know whether Pi^(Pi^(Pi^Pi)) is a whole number or not.

If a problem is fed to a supercomputer or AI, who then claims that it’s solved (proven or disproven) but yet also claims that humans would not be able to understand the proof, for some reason (such as reading the proof would take more time than a person’s lifespan). In that case, is the problem considered solved?

I didn’t know that progress had actually been made on the twin prime conjecture, also if Euler said he can’t prove it, it may be over

was literally playing bach's a minor violin concerto right before watching this

The goldbach conjecture sound weirdly very easy tho like. Knowing that all prime number after 2 are odd meaning that they could be written as (2n+1) so it would be logical to assume that the sum of any multiple of 2 which is essentially 2p. So when adding two prime number (2n+1)+(2n+1), it would just equal 4n+2 or 2(n+1). Actually nvm i understood why its really difficult, i felt like an idiot writting this

Fantastic subject matter!

<a href="#" class="seekto" data-time="707">11:47</a> Damn, the way you asked "Is Gamma Rational?" made it sound like a creepypasta from the late noughties.

I can create such problems, for example, how many different ways to place sea battle ships in n by n square exist?

Can u make a video on the abc conjecture?

how do we know that some of these so-called problems or any popular problem in general are not just necessary or natural features of a base 10 system?

<a href="#" class="seekto" data-time="392">6:32</a> Imagine someone solving what is essentially The Computer Science Problem (P = NP) just to quickly figure out whether two strings were knotted.

my math prof in the second semester put some of theese as task on our Worksheets.

Is this channel in any way associated with The Paint Explainer? Great video!

I feel like the way we do maths is not suitable for number theory, like damn how can easy problems like these be so damn hard to prove but we can prove so much things for example in higher dimensional geometry. Mindblowing...

I don't understand what "lonely" means in mathematical terms. Does it mean further than a certain distance?

<a href="#" class="seekto" data-time="415">6:55</a> prøving this would imply that P≠NP, solving a Millenium problem and giving you a lot more fame than the unknotting problem alone (in addition to a million dollars)

Unsolved Math problem that sounds Easy: are there any Lychrel numbers in Base 10 aka the 196-problem? It is quite simple to state but we simply don't know if 196, 295 etc. are Lychrel numbers.

Can I prove the knotting problem? Well, I can knot...

Great video

whats the song in the background? i know its fur elise but its different too

The definition of the Riemann Zeta function you gave is only correct for real part greater than 1. It would have been nice if you had defined it for 0 < Re(s) < 1 using the alternating zeta function.

I tought you had more subcribers, this is video is really well made. Keep up the good work

I always wondered what would happen if you gave these super simple questions (as in - super simple to STATE, not to solve) and slipped them into a high-school or lower exam.. Either they'll never get it, because it's just that difficult, if not impossible.. or, they'll see it from a completely unseen angle and make progress on it.

Ah yes as I am certain that there is no poly time algorithm for unknotting I'm quicky gonna prove P≠NP. Thanks for the tipp.

This video made it sound like Yitong Zhang proved that there's a particular number around 70million where we know there are infinitely many primes with that separation, which is not true.

Ah, that's easy! The answer is yes times maple syrup.

<a href="#" class="seekto" data-time="730">12:10</a> "Meet the Eeler-Mascheroni."