The hardest math problem to solve is currently unknown as there are many unsolved problems in various fields including number theory, algebra, and geometry.
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One of the most challenging aspects of mathematics is the sheer number of unsolved problems. While there are some notoriously difficult problems, it is difficult to definitively say which is the “hardest” to solve. As the Fields Medalist Terence Tao once said, “The difficulty of a problem is relative to what is already known.” Here are some interesting facts and examples of unsolved problems in mathematics across various fields:
- Fermat’s Last Theorem was famously unsolved for over 350 years, until it was finally proven in 1994 by Andrew Wiles.
- The Riemann Hypothesis is another famous unsolved problem, related to the distribution of prime numbers. It remains unsolved despite being one of the Clay Mathematics Institute’s Millennium Prize Problems (and the potential million-dollar prize that comes with it).
- The Collatz Conjecture, which involves iterating a simple arithmetic function on positive integers, is simple to state but has yet to be proven or disproven.
- In geometry, the Poincaré Conjecture was proven in 2002 by Grigori Perelman, who famously turned down the Fields Medal. However, the Hodge Conjecture, which relates to algebraic geometry, remains unsolved.
- Despite decades of effort, the Navier-Stokes Equations (which describe fluid dynamics) have yet to be proven to have a unique solution for all possible initial conditions.
As for a famous quote on the topic, David Hilbert once famously said in a lecture, “We must know, we shall know.” This sentiment still drives many mathematicians today as they work on unsolved problems, constantly pushing the boundaries of human knowledge.
Here is a table summarizing some of these unsolved problems:
| Field | Unsolved Problem |
|---|---|
| Number Theory | Riemann Hypothesis |
| Algebra | Hodge Conjecture |
| Geometry | Navier-Stokes Equations |
| Combinatorics | P vs. NP problem |
| Arithmetic | Langlands Program |
| Topology | Smooth Poincaré Conjecture |
| Analysis | Birch and Swinnerton-Dyer Conjecture |
Associated video
The video discusses a difficult problem on the Putnam math competition, which is a test consisting of 12 questions of varying difficulty. The problem discussed in the video is a 2D problem that can be approached in a number of ways. The video shows one way to solve the problem and emphasizes the importance of deep understanding in mathematical problem-solving.
There are other points of view available on the Internet
“There are no whole number solutions to the equation xn + yn = zn when n is greater than 2.” Otherwise known as “Fermat’s Last Theorem,” this equation was first posed by French mathematician Pierre de Fermat in 1637, and had stumped the world’s brightest minds for more than 300 years.
What Are The 7 Hardest Math Problems?
- Unknotting Problem It will be solved if you explore the simplest version of the unknotting problem.
The Continuum Hypothesis is a mathematical problem involving the concept of infinity and the size of infinite sets. It was first proposed by Georg Cantor in 1878 and has remained one of the unsolvable and hardest math problems ever since.
My favorite is the one allegedly solved by John von Neumann (among others):
Two cyclists start 30 miles apart, and begin cycling towards each other at 15 mph. The instant they begin, a fly goes from one cyclist to the other, then back to the first cyclist, and so on, always flying between the cyclists. The fly flies at a constant 30 mph. How far in total does the fly travel when the cyclists meet?
Allegedly, a student posed this problem to von Neumann, who instantly gave the correct answer
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“30 miles”, said von Neumann.
“That’s right”, said the student. “You know, Professor, most people don’t realize the cyclists meet in 1 hour, and so the fly must have flown a total of 30 miles. Instead, they think they have to add up an infinite series to get the answer.”
“But that’s what I did,” said von Neumann.