Every Unsolved Math problem that sounds Easy

the kissing number problem a broad category of problems in math are called the sphere packing problems they range from Pure math to practical stuff like figuring out how to stack many spheres in a given space like fruit at the grocery store some of these problems have Solutions but others like the kissing number problem are still tricky when a bunch of spheres are packed in some region each sphere has a kissing number which is the number of other spheres it's touching if you're touching five neighboring spheres then your kissing number is five nothing tricky a packed bunch of spheres will have an average kiss kissing number which helps mathematically describe the situation but a basic question about the kissing number stands unanswered a one-dimensional thing is a line and a two-dimensional thing is a plane for these low numbers mathematicians have proven the maximum possible kissing number for spheres of that many dimensions it's two when you're on a 1D line one sphere to your left and the other to your right there's proof of an exact number for three dimensions although that took until the 1950s Beyond three dimensions the kissing problem is mostly unsolved mathematicians have slowly whittel the possibility to fairly narrow ranges for up to 24 Dimensions with a few exactly known as you can see on this chart for larger numbers or a general form the problem is wide open there are several hurdles to a full solution including computational limitations so expect incremental progress on this problem for years to come the goldb conjecture one of the greatest Unsolved Mysteries in math is also very easy to write gold Box's conjecture is every even number greater than two is the sum of two primes you check this in your head for small numbers 18 is 13 plus + 5 and 42 is 23 + 19 computers have checked the conjecture for numbers up to some magnitude but we need proof for all natural numbers goldbach's conjecture precipitated from letters in 1742 between German mathematician Christian goldbach and legendary Swiss mathematician Leonard Oiler considered one of the greatest in math history as Oiler put it I regard it as a completely certain theorem although I cannot prove it oer may have sensed what makes this problem counterintuitively hard to solve when you look at larger numbers they have more ways of being written as sums of primes not less like how 3 + 5 is the only way to break 8 into two primes but 42 can broken into 5 + 37 11 + 31 13 + 29 and 19 + 23 so it feels like goldbox conjecture is an under statement for very large numbers still a proof of the conjecture for all numbers eludes mathematicians to this day it stands as one of the oldest open questions in all of math cat's conjecture in September 2019 news broke regarding progress on this 82-year-old question thanks to genius mathematician Terren Tao and while the story of ta's breakthrough is promising the problem isn't fully solved yet a colat conjecture is all about a function f of n which takes even numbers and cuts them in half while odd numbers get tripled and then added to one take any natural number apply F then apply F again and again you eventually land on one for every number we've ever checked the conjecture is that this is true for all natural numbers positive integers from 1 through Infinity to's recent work is a near solution to the kot's conjecture in some subtle ways but he most likely can't adapt his methods to yield a complete solution to the problem as ta subsequently explained so we might be working on it for decades longer the conjecture lives in the math discipline known as dynamical systems or the study of situations that change over time in semi-predictable ways it looks like a simple innocuous question but that's what makes it special why is such a basic question so hard to answer it serves as a benchmark for our understanding once we solve it then we can proceed to much more complicated matters the study of dynamical systems could become more robust than anyone today could imagine but we'll need to solve the colat conjecture for the subject to flourish the twin Prime conjecture together with gold box the twin Prime conjecture is the most famous in number Theory or the study of natural numbers and their properties frequently involving prime numbers since you've known these numbers since grade school stating the conjectures is easy when two primes have a difference of two they're called twin primes so 11 and 13 are twin primes as are 599 and 601 now it's a day one number Theory fact that there are infinitely many prime numbers so are there infinitely many twin primes the twin Prime conjecture says yes let's go a bit deeper the first in a pair of twin primes is with one exception always one less than a multiple of six and so the second Twin Prime is always one more than a multiple of six you can understand why if you're ready to follow a bit of he number Theory all primes after two are odd even numbers are always Zer two or four more than a multiple of six while odd numbers are always 1 3 or five more than a multiple of six well one of those three possibilities for odd numbers causes an issue if a number is three more than a multiple of six then it has a factor of three having a factor of three means a number isn't Prime with the sole exception of three itself and that's why every third odd number can't be prime how's your head after hearing this now imagine the headaches of everyone who has tried to solve this problem in the last 170 years the good news is that we've made some promising progress in the last decade mathematicians have managed to tackle closer and closer versions of the twin Prime conjecture this was their idea trouble proving there are infinitely many primes with a difference of two how about proving there are infinitely many primes with a difference of 70 million that was cleverly proven in 2013 By Yong Jong at the University of New Hampshire for the last 6 years mathematicians have been improving that number in jang's proof from Millions down to hundreds taking it down all the way to two will be the solution to the twin Prime conjecture the closest we've come given some subtle technical assumptions is 6 time will tell if the last step from 6 to 2 is right around the corner or if that last part will challenge mathematicians for decades longer the unknotting problem the simplest version of the unknotting problem has been solved so there's already some success with this story solving the full version of the problem will be an even bigger Triumph you probably haven't heard of the math subject not Theory it's taught in virtually no high schools and few colleges the idea is to try and apply formal math ideas like proofs to knots like well what you tie your shoes with for example you might know how to tie a square knot and a granny knot they have the same steps except that one twist is reversed from the square knot to the granny knot but can you prove that those knots are different well knot theorists can notot theorists Holy Grail problem was an algorithm to identify if some Tangled mess is truly knotted or if it can be disentangled to nothing the good news is that this has been accomplished several computer algorithms for this have been written in the last 20 years and some some of them even animate the process but the unknotting problem remains computational in technical terms it's known that the unknotting problem is an NE while we don't know if it's in P that roughly means that we know our algorithms are capable of unknotting knots of any complexity but that as they get more complicated it starts to take an impossibly long time for now if someone comes up with an algorithm that can unot any knot in what's called polinomial time that will put the unknotting problem fully to rest on the flip side someone could prove that isn't possible and that the unting problem's computational intensity is unavoidably profound eventually we'll find out the Enigma of pi plus e given everything we know about two of Math's most famous constants pi and E it's a bit surprising how lost we are when they're added together this mystery is all about algebraic real numbers the definition a real number is algebraic if it's the root of some polom with integer coefficients for example x^ s - 6 is a polinomial with integer coefficients since 1 and -6 are integers the roots of of x - 6 = 0 are X = < TK 6 and X = < TK 6 so that means < TK of 6 and negative square otk of 6 are algebraic numbers all rational numbers and roots of rational numbers are algebraic so it might feel like most real numbers are algebraic turns out it's actually the opposite the antonym to algebraic is Transcendental and it turns out almost all real numbers are transcendental for certain mathematical meanings of almost all so who's algebraic and who's trans transcendental the real number Pi goes back to ancient math while the number e has been around since the 17th century you've probably heard of both and you'd think we know the answer to every basic question to be asked about them right well we do know that both pi and E are transcendental but somehow it's unknown whether pi plus e is algebraic or transcendental similarly we don't know about Pi time e pi/ e and other simple combinations of them so there are incredibly basic questions about numbers we've known from Millennia that still remain Myster ious Birch and Swinton dire conjecture the Birch and Swinton dire conjecture is another of the six unsolved Millennium prize problems and it's the only other one we can remotely describe in plain English this conjecture involves the math topic known as elliptic Curves in a nutshell an elliptic curve is a special kind of function they take the unthreatening looking form y^2 = X Cub + a x + B it turns out functions like this have certain properties that cast insight into math topics like algebra and number Theory British mathematicians Brian Burch and Peter Swinton Dyer developed their conjecture in the 1960s its exact statement is very technical and has evolved over the years one of the mathematicians who has worked on refining and understanding this conjecture is Andrew WS who's famous for proving Fermat's Last Theorem reman hypothesis today's mathematicians would probably agree that the reman hypothesis is the most significant open problem in all of math it's one of the seven Millennium prize problems with 1 million reward for its solution it has implications deep into various branches of math but it's also simple enough that we can explain the basic idea right here there is a function called the reman zeta function for each s this function gives an infinite sum which takes some basic calculus to approach for even the simplest values of s for example if s equal 2 then Zeta of s is the well-known series which strangely adds up to exactly pi^ 2 / 6 when s is a complex number one that looks like a + b * I using the imag inary number I finding Zeta of s gets tricky so tricky in fact that it's become the ultimate math question specifically the rean hypothesis is about when Zeta of s equals z the official statement is every non-trivial zero of the rean zeta function has real part 1/2 on the plane of complex numbers this means the function has a certain Behavior along a special vertical line the hypothesis is that the behavior continues along that line infinitely the hypothesis and the zeta function come from German mathematician burnhard reman who described them in 1859 reman developed them while studying prime numbers and their distribution our understanding of prime numbers has flourished in the 160 years since and rean would never have imagined the power of supercomputers but lacking a solution to the rean hypothesis is a major setback if the rean hypothesis were solved tomorrow it would unlock an avalanche of further progress it would be huge news throughout the subjects of number Theory and Analysis until then the rean hypothesis remains one of the largest dams to the river of math research the lonely Runner conjecture assume there are some kids running around a circular track each kid runs at a different speed now if a kid gets far enough away from all the other kids we say that kid is lonely this idea is called the lonely Runner conjecture it says that every kid running will become lonely at some point if there's only one kid running it's easy because they're always lonely since no one else is there with two kids it's all so simple we just watch how fast they run by pretending one kid is standing still we can see when the other kid gets lonely for 3 4 5 6 and seven kids smart people have already figured out that each kid will become lonely at some point but for more than seven kids we're still not sure there have been some guesses and ideas like looking at really big groups of kids running at different speeds but the problem hasn't been completely solved yet is gamma rational this is another easy to write problem but hard to solve all you need to recall is the definition of rational numbers rational numbers can be written in the form p over Q where p and Q are integers so 42 and13 are rational while pi and theare < TK of two are not it's a very basic property so you'd think we can easily tell when a number is rational or not right meet the eer mascheroni it's a real number approximately 05772 with a closed form that's not terribly ugly the Sleek way of putting words to those symbols is gamma is the limit of the difference of the harmonic series and the natural log so it's a combination of two very well understood mathematical objects it has other neat closed forms and appears in hundreds of formulas but somehow we don't even know if gamma is rational people have calculated it to half a trillion digits yet nobody can prove if it's rational or not the popular prediction is that gamma is irrational along with our previous example pi plus e we have another question of a simple property for a well-known number and we can't even answer it that's it for today if you enjoyed exploring these Mysteries with me please like And subscribe your support means a lot

Channel: ThoughtThrill

Views: 390,133

Keywords:

Id: WxXaE4zVwx0

Length: 12min 53sec (773 seconds)

Published: Wed Jun 19 2024