Despite recent breakthroughs like a supercomputer cracking the Sum of Three Cubes problem after 65 years, mathematicians continue to grapple with centuries-old challenges, reminding us that even the toughest math problems may eventually yield to human perseverance, or perhaps not.
Sources:
Notes from the following web article: https://www.popularmechanics.com/science/math/g29251596/impossible-math-problems/
For all of the recent strides we’ve made in the math world—like a supercomputer finally solving the Sum of Three Cubes problem that puzzled mathematicians for 65 years—we’re forever crunching calculations in pursuit of deeper numerical knowledge.
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The Sum of three cubes problem, also known as the Fermat’s Last Theorem for the special case of exponent 3, asks whether every positive integer can be expressed as the sum of three cubes. In other words, it seeks to find solutions to the equation x^3 + y^3 + z^3 = n for any given positive integer n. This problem has a long history and has been extensively studied by mathematicians. In 1954, L.J. Lander and T.R. Parkin found a solution for n=33 that involved very large numbers. Since then, various other solutions have been discovered, including n=42 and n=114. However, there is still no general formula or algorithm known that can express any positive integer as the sum of three cubes. The problem remains an open question in mathematics and is considered one of the most challenging unsolved problems in number theory.
More about the problems and solutions of the three cubes problem on: https://news.mit.edu/2021/solution-3-sum-cubes-puzzle-0311
The top 10 hardest problems known to humanity are:
1 - The Collatz Conjecture
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The Collatz Conjecture is a famous unsolved problem in mathematics that was first proposed by German mathematician Lothar Collatz in 1937. The conjecture is quite simple to state but remains unproven to this day. The conjecture starts with a positive integer, and repeatedly applies the following rules:
- If the number is even, divide it by 2.
- If the number is odd, multiply it by 3 and add 1.
The process is then repeated with the resulting number, and it continues until reaching the number 1. The conjecture states that no matter what positive integer you start with, eventually you will always reach the number 1. For example, if we start with the number 6: 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 Although this pattern holds true for many numbers that have been tested, no proof has been found to demonstrate that it holds true for all positive integers. The Collatz Conjecture remains an intriguing and unsolved problem in mathematics.
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 onto much more complicated matters.
2 - Goldbach’s Conjecture
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Goldbach’s conjecture is a famous unsolved problem in number theory. It states that every even integer greater than 2 can be expressed as the sum of two prime numbers. For example, 4 can be expressed as 2 + 2, 6 as 3 + 3, 8 as 3 + 5, and so on. Despite being proposed by the German mathematician Christian Goldbach in a letter to Euler in 1742, no counterexamples have been found, and it has been checked for even numbers up to very large limits without any exceptions. However, a proof for this conjecture remains elusive, and it remains an open problem in mathematics.
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 Goldbach’s Conjecture is an understatement 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.
3 - The Twin Prime Conjecture
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The twin prime conjecture is an unsolved problem in number theory that suggests there are infinitely many pairs of prime numbers that differ by 2. These pairs are called twin primes. For example, (3, 5), (11, 13), and (17, 19) are all twin prime pairs. The conjecture states that no matter how far we go in the sequence of prime numbers, there will always be another pair of twin primes. Despite extensive computational evidence supporting this conjecture, a proof has not yet been found.
Now, it’s a Day 1 Number Theory fact that there are infinitely many prime numbers. So, are there infinitely many twin primes? The Twin Prime Conjecture says yes.
There was already proved (with some subtle technical assumptions) infinite prime numbers with a difference of six. 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.
4 - The Riemann Hypothesis
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The Riemann hypothesis is one of the most famous unsolved problems in mathematics. It is named after the German mathematician Bernhard Riemann, who first stated it in 1859. The hypothesis deals with the distribution of prime numbers, which are numbers that can only be divided by 1 and themselves. In simple terms, the Riemann hypothesis states that all non-trivial zeros of the Riemann zeta function have a real part equal to 1/2. The Riemann zeta function is a mathematical function that plays a crucial role in number theory and has connections to many other areas of mathematics. If the hypothesis is true, it would have far-reaching implications for various branches of mathematics, including prime number theory and the understanding of how prime numbers are distributed among all integers. It would also provide insights into other mathematical problems related to complex analysis and number theory. Although the Riemann hypothesis has been extensively studied for over a century, no proof or disproof has been found yet. It remains an open question that continues to intrigue and challenge mathematicians around the world.
Today’s mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math.
There is a function, called the Riemann zeta function, written in the image above. 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=2, then 𝜁(s) is the well-known series 1 + 1/4 + 1/9 + 1/16 + …, which strangely adds up to exactly 𝜋²/6. When s is a complex number—one that looks like a+b𝑖, using the imaginary number 𝑖—finding 𝜁(s) gets tricky.
So tricky, in fact, that it’s become the ultimate math question. Specifically, the Riemann Hypothesis is about when 𝜁(s)=0; the official statement is, “Every nontrivial zero of the Riemann 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.
If the Riemann 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 Riemann Hypothesis remains one of the largest dams to the river of math research.
5 - The Birch and Swinnerton-Dyer Conjecture
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The Birch and Swinnerton-Dyer Conjecture is a famous unsolved problem in mathematics. It was proposed by mathematicians Bryan Birch and Peter Swinnerton-Dyer in 1965. The conjecture deals with elliptic curves, which are a type of mathematical object described by an equation of the form y^2 = x^3 + ax + b. The conjecture states that for any given elliptic curve, there is a relationship between the number of rational points on the curve and certain properties of its associated mathematical function called the L-function. In particular, if the L-function has a non-zero value at a certain point, then there should be infinitely many rational points on the curve. On the other hand, if the L-function is zero at that point, then there should be only finitely many rational points. If proven true, the Birch and Swinnerton-Dyer Conjecture would have profound implications for number theory and algebraic geometry. It would provide a deeper understanding of elliptic curves and their behavior, as well as potentially solving other related problems. Despite significant progress made by mathematicians over the years, including partial results known as “rank predictions,” a complete proof or disproof of the conjecture remains elusive. It is considered one of the most important open problems in mathematics today.
One of the greatest achievements in 20th-century math was the solution to Fermat’s Last Theorem. Sir Andrew Wiles solved it using Elliptic Curves. So, you could call this a very powerful new branch of math. More about this on: https://theconversation.com/proving-fermats-last-theorem-2-mathematicians-explain-how-building-bridges-within-the-discipline-helped-solve-a-centuries-old-mystery-207968
British mathematicians Bryan Birch and Peter Swinnerton-Dyer developed their conjecture in the 1960s. Its exact statement is very technical, and has evolved over the years. One of the main stewards of this evolution has been none other than Wiles. To see its current status and complexity, check out this famous update by Wells in 2006.
6 - The Kissing Number Problem
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The kissing number problem is a mathematical problem that asks the maximum number of non-overlapping spheres that can touch a central sphere of the same size in n-dimensional space. In other words, it seeks to determine the highest possible number of spheres that can be arranged around a central sphere, all touching it without overlapping. The problem was named “kissing number” because it can be visualized as finding the maximum number of spheres that can “kiss” or touch a central sphere. It has applications in fields such as coding theory, crystallography, and physics. The kissing number problem is a challenging problem in higher dimensions because intuitive geometric insights from two or three dimensions do not necessarily apply. The solution depends on the dimensionality of space, and it has been solved for certain dimensions. However, for higher dimensions, the exact solution remains unknown.
A broad category of problems in math are called the Sphere Packing Problems. They range from pure math to practical applications, generally putting math terminology to the idea of stacking many spheres in a given space, like fruit at the grocery store.
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 6 neighboring spheres, then your kissing number is 6. Nothing tricky. A packed bunch of spheres will have an average kissing number, which helps mathematically describe the situation.
A 1-dimensional thing is a line, and 2-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 2 when you’re on a 1-D line—one sphere to your left and the other to your right. There’s proof of an exact number for 3 dimensions, although that took until the 1950s.
Beyond 3 dimensions, the Kissing Problem is mostly unsolved. Mathematicians have slowly whittled the possibilities to fairly narrow ranges for up to 24 dimensions, with a few exactly known as seen in figure below. 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.
Source: https://en.wikipedia.org/wiki/Kissing_number#/media/File:Kissing_growth.svg
7 - The Unknotting Problem
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The unknotting problem, also known as the word problem for the unknot, is a problem in knot theory. It involves determining whether a given knot diagram represents the trivial knot (i.e., the unknot) or a nontrivial knot. In other words, it asks whether it is possible to transform a given knot into the simple loop of the unknot through a sequence of Reidemeister moves, which are local operations that do not change the underlying knot type. The unknotting problem is a fundamental question in knot theory and remains an active area of research.
You probably haven’t heard of the math subject Knot 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.
Knot 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 cool news is that this has been accomplished! Several computer algorithms for this have been written in the last 20 years, and some of them even animate the process.
But the Unknotting Problem remains computational. In technical terms, it’s known that the Unknotting Problem is in NP, 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 unknot any knot in what’s called polynomial time, that will put the Unknotting Problem fully to rest. On the flip side, someone could prove that isn’t possible, and that the Unknotting Problem’s computational intensity is unavoidably profound. Eventually, we’ll find out.
8 - The Large Cardinal Project
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The large cardinal project is a research program in set theory that aims to study and classify large cardinal axioms. Large cardinals are certain types of infinite cardinal numbers that possess strong properties. These axioms have been found to have significant implications in various areas of mathematics, such as logic, topology, and algebra. The project involves investigating the consistency and structural properties of different large cardinal axioms, as well as their relationship to other mathematical concepts. It also explores the hierarchy of large cardinals and their role in establishing the foundations of mathematics. The study of large cardinals has provided insights into the nature of infinity and the limitations of various mathematical systems. It has also led to the development of new techniques and tools in set theory and related fields. Overall, the large cardinal project seeks to deepen our understanding of the foundational principles of mathematics by exploring these powerful and intriguing mathematical objects.
In the late 19th century, a German mathematician named Georg Cantor figured out that infinity comes in different sizes. Some infinite sets truly have more elements than others in a deep mathematical way, and Cantor proved it.
There is the first infinite size, the smallest infinity, which gets denoted ℵ₀. It’s the size of the set of natural numbers, so that gets written |ℕ| = ℵ₀. The major example Cantor proved is that the set of real numbers is bigger, written |ℝ| > ℵ₀. But the reals aren’t that big; we’re just getting started on the infinite sizes.
For the really big stuff, mathematicians keep discovering larger and larger sizes, or what we call Large Cardinals. It’s a process of pure math that goes like this: Someone says, “I thought of a definition for a cardinal, and I can prove this cardinal is bigger than all the known cardinals.” Then, if their proof is good, that’s the new largest known cardinal. Until someone else comes up with a larger one. There’s now even a wiki of known large cardinals. So, will this ever end? The answer is broadly yes, although it gets very complicated.
In some senses, the top of the large cardinal hierarchy is in sight. Some theorems have been proven, which impose a sort of ceiling on the possibilities for large cardinals. But many open questions remain, and new cardinals have been nailed down as recently as 2019. It’s very possible we will be discovering more for decades to come. Hopefully we’ll eventually have a comprehensive list of all large cardinals.
9 - What’s the Deal with 𝜋+e?
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𝜋+e refers to the sum of the mathematical constants 𝜋 (pi) and e (Euler’s number). It is an interesting and mysterious mathematical expression that has captured the curiosity of mathematicians. The value of 𝜋 is approximately 3.14159, and e is approximately 2.71828. When you add these two numbers together, you get a result known as 𝜋+e. The significance of 𝜋+e lies in its connection to various areas of mathematics. It appears in equations related to calculus, complex analysis, number theory, and more. The sum has been studied extensively by mathematicians, but its exact properties and behavior are not fully understood. One fascinating aspect of 𝜋+e is its irrationality. Both 𝜋 and e are irrational numbers, meaning they cannot be expressed as a fraction. When you add two irrational numbers together, the result is also an irrational number. This means that 𝜋+e has an infinite non-repeating decimal expansion without any pattern. Despite its mystery, 𝜋+e does not have any direct practical applications in everyday life. However, it maintains a special place in the world of mathematics as an intriguing mathematical expression that continues to fascinate and challenge mathematicians.
Given everything we know about two of math’s most famous constants, 𝜋 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 polynomial with integer coefficients. For example, x²-6 is a polynomial with integer coefficients, since 1 and -6 are integers. The roots of x²-6=0 are x=√6 and x=-√6, so that means √6 and -√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 transcendental?
Well, we do know that both 𝜋 and e are transcendental. But somehow it’s unknown whether 𝜋+e is algebraic or transcendental. Similarly, we don’t know about 𝜋e, 𝜋/e, and other simple combinations of them. So there are incredibly basic questions about numbers we’ve known for millennia that still remain mysterious.
10 - Is 𝛾 Rational?
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The famous math problem “Is 𝛾 Rational?” refers to the question of whether the Euler-Mascheroni constant (𝛾) is a rational number or an irrational number. The Euler-Mascheroni constant is a mathematical constant that appears in many areas of mathematics, particularly in number theory and calculus. It is defined as the limiting difference between the harmonic series and the natural logarithm of the natural numbers. As of now, it is still an open question whether 𝛾 is rational or irrational. Extensive computational evidence suggests that 𝛾 is irrational, but a proof has not yet been established. Proving the rationality or irrationality of 𝛾 remains an active area of research in mathematics.
Rational numbers can be written in the form p/q, where p and q are integers. So, 42 and -11/3 are rational, while 𝜋 and √2 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 Euler-Mascheroni constant 𝛾, which is a lowercase Greek gamma. It’s a real number, approximately 0.5772, with a closed form that’s not terribly ugly; it looks like the image above.
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 𝛾 is rational. We’ve calculated it to half a trillion digits, yet nobody can prove if it’s rational or not. The popular prediction is that 𝛾 is irrational. Along with our previous example 𝜋+e, we have another question of a simple property for a well-known number, and we can’t even answer it.