The Most Difficult Unsolved Mathematical Problem, Ranked

Voting rules: Choose the unsolved mathematical problem you think is the most difficult!

Updated on May 3, 2024 06:26

In the realm of mathematics, there are challenges that have baffled even the brightest minds for centuries. These problems, profound in their complexity, are not just puzzles to be solved; they are gatekeepers of deeper understanding and advanced scientific progress. Ranking these enigmas helps to highlight where collective human ingenuity has been stumped, drawing more attention and resources to these critical areas. By participating in voting for the toughest unsolved mathematical problems, users contribute to a global discussion about the priorities in mathematical research. This consensus on what the academic community and enthusiasts consider the most pressing puzzles can inspire new efforts and collaborations. It's a dynamic way for everyone to engage with the frontiers of mathematical exploration and potentially stimulate breakthroughs that everyone, from scholars to hobbyists, can celebrate.

What Is the Most Difficult Unsolved Mathematical Problem?

1

98
votes
Riemann Hypothesis

Bernhard Riemann

This problem is considered the most difficult unsolved problem in mathematics since it is one of the seven Millennium Prize Problems that the Clay Mathematics Institute has offered a million-dollar reward for its solution. It deals with the distribution of prime numbers and their relationship with the zeros of the Riemann zeta function.
The Riemann Hypothesis is one of the most significant unsolved problems in mathematics. It is a conjecture made by German mathematician Bernhard Riemann in 1859, regarding the distribution of prime numbers. Essentially, it provides information about the behavior of the Riemann zeta function, which is a complex-valued function that holds great importance in number theory. The Riemann Hypothesis suggests that all non-trivial zeros of the zeta function have a real part of 1/2.

Conjecture Type: Unsolved

Field: Number Theory

Conjectured Date: 1859

Importance: Critical

Significance: Determines prime numbers distribution
Riemann Hypothesis in other rankings
- Position 2 of 10 in Most difficult math equation
- Position 2 of 10 in Most difficult equation
- Position 3 of 10 in Most advanced mathematical proof used in reasoning
- Position 4 of 10 in Most difficult equation to solve
2

28
votes
Birch and Swinnerton-Dyer Conjecture

This conjecture is another Millennium Prize Problem that has to do with elliptic curves and their associated L-functions. It seeks to determine whether there is a simple formula that can predict the rank of an elliptic curve, which would have implications for many areas of mathematics.
The Birch and Swinnerton-Dyer Conjecture is one of the unsolved problems in mathematics that relates to elliptic curves, which are a type of mathematical object with fascinating properties. It was proposed by Bryan Birch and Peter Swinnerton-Dyer in 1965. The conjecture posits that there is a fundamental connection between the arithmetic properties of elliptic curves and the behavior of their associated L-functions. If the L-function of an elliptic curve has a specific value at a certain critical point, then the curve has an infinite number of rational points. On the other hand, if the L-function has a specific value of zero at that point, then the curve has only finitely many rational points. The conjecture has wide-ranging implications in number theory and algebraic geometry.

Year Proposed: 1965

Type of Problem: Unsolved Problem

Mathematical Field: Number Theory and Algebraic Geometry

Subject Area: Elliptic Curves

Connection: Arithmetic properties of elliptic curves and their L-functions
Birch and Swinnerton-Dyer Conjecture in other rankings
- Position 4 of 10 in Most difficult math question
- Position 4 of 10 in Most advanced math problem
- Position 7 of 10 in Most difficult equation to solve
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18
votes
Hodge Conjecture

William Vallance Douglas Hodge

This conjecture is about algebraic geometry and seeks to understand the relationship between the topology of a complex projective variety and its algebraic structure. It is also one of the Millennium Prize Problems.
The Hodge Conjecture is a famous conjecture in algebraic geometry. It deals with the relationship between two important mathematical structures, algebraic cycles and cohomology theory. The conjecture states that for a smooth projective algebraic variety, there exists a natural decomposition of the cohomology into pieces called Hodge classes, which are algebraic cycles. These algebraic cycles can be understood in terms of the geometry of the variety. The conjecture further asserts that any Hodge class can be represented by a rational linear combination of algebraic cycles.

Field: Algebraic geometry

Status: Unsolved

Conjectured: During the 1950s

Importance: One of the seven Millennium Prize Problems

Impact: Reveals deep connections between algebraic cycles and cohomology theory
Hodge Conjecture in other rankings
- Position 7 of 10 in Most advanced mathematical proof used in reasoning
- Position 7 of 10 in Most advanced math problem
4

22
votes
Navier-Stokes Equations

These equations describe the motion of fluids and have been used to model everything from weather patterns to blood flow. However, a complete mathematical understanding of the equations has yet to be achieved, making them one of the most famous unsolved problems in physics and mathematics.
The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluid substances such as liquids and gases.

Equations: The equations consist of the continuity equation and the momentum equations.

Continuity equation: Describes the conservation of mass for an incompressible fluid.

Momentum equations: Describe the conservation of momentum for a fluid, taking into account pressure and viscous forces.

Nonlinear nature: The equations are nonlinear, making them difficult to solve analytically.

Important in fluid dynamics: They provide a mathematical description of the behavior of fluids in various scenarios.
Navier-Stokes Equations in other rankings
- Position 1 of 10 in Most difficult math equation
- Position 1 of 10 in Most difficult equation
- Position 3 of 10 in Most difficult math question
- Position 5 of 10 in Most advanced math problem
- Position 8 of 10 in Most advanced mathematical proof used in reasoning
5

12
votes
Yang-Mills Existence and Mass Gap

This is another Millennium Prize Problem that seeks to understand the behavior of quantum field theories. Specifically, it asks whether or not there is a mass gap in the theory of quantum chromodynamics, which describes the interactions between quarks and gluons.
The Yang-Mills Existence and Mass Gap problem is a fundamental question in theoretical physics and mathematics. It is related to the behavior of quantum chromodynamics (QCD), which describes the strong force that binds quarks together in atomic nuclei and governs interactions between elementary particles.

Problem Type: Mathematical physics problem

Nature: Unsolved problem

Field: Theoretical physics and mathematics

Main Question: Does the Yang-Mills theory have a mass gap and exist in four dimensions?

Yang-Mills Theory: A gauge theory that describes interactions among elementary particles based on the symmetry group SU(N)
Yang-Mills Existence and Mass Gap in other rankings
- Position 3 of 10 in Most advanced math problem
6

7
votes
P versus NP

Stephen Cook and Leonid Levin

This problem is about computational complexity and asks whether or not every problem that can be verified in polynomial time can also be solved in polynomial time. It has many practical implications for computer science and cryptography.
P versus NP is a fundamental problem in computer science and mathematics that deals with the efficiency of algorithms. It seeks to determine if every problem with a feasible solution also has an efficient solution. In simple terms, it asks whether it is easier to verify a solution (P) than to find the solution itself (NP). It is considered one of the most significant unsolved problems in computer science.

Importance: One of the most important unsolved problems in computer science

Complexity theory: P is the complexity class that represents problems solvable in polynomial time, while NP represents problems whose solutions can be verified in polynomial time

Significance: If P = NP, it would imply that many difficult problems can be solved efficiently, potentially transforming fields like cryptography, optimization, and artificial intelligence

Millennium Prize Problem: P versus NP is one of the seven unsolved problems in mathematics designated by the Clay Mathematics Institute as a Millennium Prize Problem

NP-Complete problems: If P = NP, all NP problems are also in P, including the NP-Complete problems which are considered to be the hardest problems within NP
P versus NP in other rankings
- Position 6 of 10 in Most difficult equation to solve
7

8
votes
The Collatz Conjecture

This conjecture is a simple problem that has eluded mathematicians for decades. It asks whether or not every positive integer will eventually reach 1 when subjected to a certain iterative process. Despite its simplicity, no one has been able to prove or disprove the conjecture.
The Collatz Conjecture is an unsolved mathematical problem that involves iteratively applying a sequence of operations to a positive integer. The conjecture states that regardless of the starting value, this sequence will always eventually reach the value 1.

Operation: If the current number is even, divide it by 2. If the current number is odd, multiply it by 3 and add 1.

Iterative Process: Repeat the operation on the resulting number.

Expected Convergence: The conjecture suggests that for any positive integer input, the sequence will eventually reach the value 1.

Open Problem: The conjecture remains unproven, despite extensive computational verification for many starting values.

Lothar Collatz Prize: In 2018, the Collatz Conjecture gained attention with the announcement of a $1 million prize for its solution.
The Collatz Conjecture in other rankings
- Position 7 of 10 in Most difficult math question
- Position 9 of 10 in Most difficult equation to solve
8

3
votes
Twin Prime Conjecture

Alphonse de Polignac

This conjecture is about prime numbers and asks whether or not there are infinitely many pairs of primes that differ by 2 (e.g., 3 and 5, 11 and 13, etc.). Although this problem has been around for centuries, no one has been able to prove or disprove it definitively.
The Twin Prime Conjecture suggests that there are infinitely many pairs of prime numbers that differ by exactly two.

Status: Unproven

Formulated: Unknown (pre-1846)

Importance: High

Difficulty: Open

Field: Number theory
9

4
votes
Adam Cunningham and John Ringland · CC BY-SA 3.0

Goldbach's Conjecture

Christian Goldbach

This conjecture is about even numbers and asks whether or not every even number greater than 2 can be expressed as the sum of two prime numbers. Despite being around for over 250 years, no one has been able to prove or disprove the conjecture.
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 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, and so on. The conjecture was first proposed by the German mathematician Christian Goldbach in a letter to the Swiss mathematician Euler in 1742.

Conjecture: Every even integer greater than 2 can be expressed as the sum of two prime numbers.

First Proposed: 1742

Number Theory Problem: Yes

Unsolved: Yes

Even Numbers: Yes
10

7
votes
The Hierarchy Problem

Edward Witten

This is a problem in particle physics that seeks to understand why gravity is so much weaker than the other fundamental forces of nature. While there are many proposed solutions, none have been definitively proven or accepted by the scientific community.
The Hierarchy Problem is a problem in theoretical physics that arises from the large discrepancy between the weak force and the gravitational force in terms of their strength. It refers to the question of why gravity is so much weaker than the other fundamental forces, even though it acts on the same particles.

Problem Type: Theoretical physics problem

Origin: Late 20th century

Discrepancy: Large strength difference between weak and gravitational forces

Fundamental Forces: Comparing gravity with other fundamental forces

Particle Interaction: Gravity acting on the same particles

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Discussion

Ranking factors for difficult unsolved mathematical problem

Historical significance

Some problems gain importance due to their history and connection with renowned mathematicians or mathematical breakthroughs. The longer a problem remains unsolved, the more challenging it is perceived to be.
Interconnectedness

Many mathematical problems are intertwined with other problems, either directly or conceptually. The more connected a problem is to other unsolved issues, the more difficult it becomes to isolate it and find a solution.
Number of failed attempts

The number of attempts mathematicians have made to solve a problem and the various strategies that have been employed can also provide insight into its difficulty.
Prizes or recognition

Some problems have monetary rewards or other forms of recognition associated with them, indicating their perceived difficulty and importance within the mathematical community.
Existence of partial solutions or progress

The degree to which a problem has been partially solved or progress made towards a solution can be a factor in ranking its difficulty. A problem with little or no progress over time is considered to be more difficult.
Accessibility and applicability

Problems that require specialized knowledge or are limited to a specific subfield may be considered more difficult than those that involve more accessible or widely applicable concepts.
Uniqueness and novelty

Some unsolved problems are difficult because they are unique or involve novel ideas within their field; these problems may require a mathematician to think in entirely new ways to arrive at a solution.
Dependence on conjectures

A problem's difficulty may be related to its dependence on unproven conjectures or assumptions. If a problem relies heavily on conjectural results, it can be considered more difficult, since solving it may require proving the conjectures first.

About this ranking

This is a community-based ranking of the most difficult unsolved mathematical problem. We do our best to provide fair voting, but it is not intended to be exhaustive. So if you notice something or puzzle is missing, feel free to help improve the ranking!

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More information on most difficult unsolved mathematical problem

Mathematics is a fascinating field that has challenged human intellect for centuries. It has given us some of the most important discoveries and inventions that we use today. However, there are still some unsolved problems in mathematics that have puzzled mathematicians for years. These problems are not only difficult to solve, but they also have far-reaching implications in various fields such as computer science, physics, and cryptography. One such problem is the P versus NP problem, which asks whether every problem with a yes or no answer can be solved efficiently by a computer. Another famous problem is the Riemann Hypothesis, which deals with the distribution of prime numbers and has implications in cryptography and data encryption. The Birch and Swinnerton-Dyer Conjecture is another unsolved problem that deals with elliptic curves and has implications in number theory. These problems and many others like them are still unsolved, despite the best efforts of mathematicians around the world. However, the pursuit of solving these problems continues to inspire new breakthroughs in mathematics and other fields.