Advanced math problems are undeniably the most difficult, with many of them remaining unsolved. We are always crunching calculations in pursuit of deeper numerical knowledge despite all of the advances we have made in the math world recently. For example, a supercomputer that solved the Sum of Three Cubes problem vexed mathematicians for 65 years.

Math problems have stumped us for centuries, and while some brain teasers may appear impossible to solve by human or machine, someone might inevitably solve them… or not. Only time will tell. Nonetheless, read on to discover if you’re the math brain who can crack the top unsolvable math problems of all time.

1. Separatrix Separation

A moving pendulum can swing from side to side or spin in a continuous circle. The separatrix is the point at which one type of motion transitions to another, and it can be calculated in most simple situations. However, when the pendulum is prodded at a nearly constant rate, the math falls apart.

Can such a separatrix be described by an equation?1

1. The Collatz Conjecture

The Collatz conjecture focuses primarily on sequences beginning with any positive integer.

It states that if we start with any positive number, such as n, then each term is obtained as follows:

• If the previous term is even, the next term is one half of the previous term, that is n/2.
• If the previous term is odd, the next term is 3 times the previous term plus 1, that is 3n+1.5

Thanks to prolific mathematician Terence Tao, news about progress on this 82-year-old question recently broke. While the story of Tao’s breakthrough is encouraging, the problem has not yet been fully resolved. In some ways, Tao’s recent work comes close to solving the Collatz Conjecture. However, as Tao later explained, he is unlikely to be able to adapt his methods in order to produce a complete solution to the problem.This may result in mathematicians working on it for decades to come.2

1. Riemann Hypothesis

Riemann’s Hypothesis is one of the most important open problems in all of mathematics. It has far-reaching implications in a variety of fields of math, but it’s also straightforward. According to the Riemann hypothesis, “all interesting solutions of the equation

ζ(s) = 0

lie on a certain vertical straight line.”3

It’s been checked for 10,000,000,000,000 solutions, proving that it is true that every interesting solution will shed light on many of the mysteries surrounding prime number distribution. It’s so difficult that it’s become the ultimate math problem, and it’s also one of the Millennium Prize Problems with a \$1 million prize for solving it.3

1. Navier-Stokes Equations

The Navier-Stokes equations are another Millennium Prize Problems with a \$1 million prize. The equations, which were developed in 1822, are used to describe the motion of viscous fluids such as air passing over an aircraft wing or water flowing from a faucet. However, there are times when it’s difficult to tell whether the equations fail or give no answer at all. Many mathematicians have attempted, but failed to solve the problem.1

1. Birch and Swinnerton-Dyer Conjecture

The problem of describing all solutions in whole numbers x,y,z to algebraic equations such as   x2 + y2 = z2 has always fascinated mathematicians. Euclid provided the complete solution for that equation, but this becomes extremely difficult for more complicated equations, and there is no general method for determining when such equations have a whole-number solution. The conjecture was developed in the 1960s by British mathematicians Bryan Birch and Peter Swinnerton-Dyer, and its exact formulation is very technical and has evolved over time.4

References:

1. 5 of the world’s toughest unsolved maths problems | New Scientist. (n.d.). Retrieved June 8, 2022, from https://www.newscientist.com/article/2193080-5-of-the-worlds-toughest-unsolved-maths-problems/
2. The Simple Math Problem We Still Can’t Solve | Quanta Magazine. (n.d.). Retrieved June 8, 2022, from https://www.quantamagazine.org/why-mathematicians-still-cant-solve-the-collatz-conjecture-20200922/
3. Riemann Hypothesis | Clay Mathematics Institute. (n.d.). Retrieved June 8, 2022, from https://www.claymath.org/millennium-problems/riemann-hypothesis
4. Birch and Swinnerton-Dyer Conjecture | Clay Mathematics Institute. (n.d.). Retrieved June 8, 2022, from https://www.claymath.org/millennium-problems/birch-and-swinnerton-dyer-conjecture
5. Collatz Conjecture: The Simplest Conjecture In Mathematics That No One Has Ever Proved. (n.d.). Retrieved June 9, 2022, from https://www.secretsofuniverse.in/collatz-conjecture/