The Most Difficult Math Question, Ranked

Voting rules: Choose the math question you think is the most difficult!

Updated on Apr 30, 2024 06:24

In the world of mathematics, as students and educators alike delve deeper into complex theories and problems, distinguishing the challenging from the complex becomes a fascinating endeavor. This distinction helps focus learning efforts, prepares students for advanced topics, and aids instructors in curriculum development. By identifying and ranking the most difficult math questions, we provide a valuable resource for both academic growth and intellectual challenge. This list serves as a community-driven guide where enthusiasts and scholars contribute their insights by voting on the complexity and intricacy of various math questions. It's not just a list, but a dynamic reflection of collective experience and knowledge. Through your participation, you directly influence which problems are highlighted as the most challenging, helping others to gauge the mountains worth climbing in the vast landscape of mathematics.

What Is the Most Difficult Math Question?

1

39
votes
The Riemann Hypothesis

Bernhard Riemann

This is considered one of the most difficult and unsolved problems in mathematics. It deals with the distribution of prime numbers and has far-reaching implications in fields such as cryptography and number theory.
The Riemann Hypothesis is one of the most significant unsolved problems in mathematics. It was formulated by the German mathematician Bernhard Riemann in 1859 and is concerned with the distribution of prime numbers. This hypothesis states that all non-trivial zeros of the Riemann zeta function have a real part equal to 1/2.

Formulated: 1859

Subject: Distribution of prime numbers

Hypothesis: All non-trivial zeros have real part equal to 1/2

Zeta Function: Riemann zeta function

Number Theory: Important problem in number theory
The Riemann Hypothesis in other rankings
- Position 1 of 10 in Most advanced math problem
- Position 2 of 10 in Most difficult Calculus problem
2

40
votes
P vs NP

Stephen Cook

This is another unsolved problem in computer science and mathematics, which deals with the complexity of algorithms. Essentially, it asks whether every problem that can be verified by a computer can also be solved by a computer in a reasonable amount of time.
P vs NP is an unsolved problem in computer science that deals with the classification of computational problems. It asks whether every problem for which a solution can be verified quickly can also be solved quickly. The problem belongs to the field of computational complexity theory.

Problem type: Decision problem

Problem status: Unsolved

Problem difficulty: Open problem

Problem domain: Computational complexity theory

Importance: One of the most important problems in computer science
3

26
votes
Navier-Stokes Equations

These are partial differential equations that describe the motion of fluids such as air and water. They are notoriously difficult to solve and have been a subject of research for over 150 years.
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 4 of 10 in Most difficult unsolved mathematical problem
- Position 5 of 10 in Most advanced math problem
- Position 8 of 10 in Most advanced mathematical proof used in reasoning
4

21
votes
Birch and Swinnerton-Dyer Conjecture

This is a conjecture in number theory that connects elliptic curves to the distribution of prime numbers. It has far-reaching implications in cryptography and is still unsolved.
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 2 of 10 in Most difficult unsolved mathematical problem
- Position 4 of 10 in Most advanced math problem
- Position 7 of 10 in Most difficult equation to solve
5

16
votes
Fermat's Last Theorem

Pierre de Fermat

This was a problem that stumped mathematicians for over 350 years until it was finally solved in 1994. The theorem states that there are no whole number solutions to the equation a^n + b^n = c^n for n greater than 2.
Fermat's Last Theorem is a famous mathematical statement that was first proposed by Pierre de Fermat in 1637. It states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.

Conjecture: a^n + b^n = c^n has no integer solutions for n > 2

Proof: The proof was completed by Andrew Wiles with assistance from Richard Taylor

Year of Proof: 1994

Field: Number Theory

Significance: One of the most famous and long-standing unsolved problems in mathematics
Fermat's Last Theorem in other rankings
- Position 1 of 10 in Most advanced mathematical proof used in reasoning
- Position 3 of 10 in Most difficult equation to solve
- Position 4 of 10 in Most difficult equation
- Position 4 of 10 in Most difficult math equation
6

10
votes
The Four Color Theorem

This theorem states that any map can be colored with no more than four colors, such that no two adjacent regions have the same color. It was first proposed in the 1850s but was not proven until 1976.
The Four Color Theorem is a famous problem in graph theory which states that any map can be colored using at most four colors in such a way that no two adjacent regions have the same color. An adjacent region is defined as two regions that share a common boundary, not just a point. This problem was first stated in 1852 and was eventually proven in 1976 after a long and complex mathematical argument.

Problem type: Graph theory

Year stated: 1852

Year proven: 1976

Number of colors: 4

Definition of adjacent regions: Regions that share a common boundary
The Four Color Theorem in other rankings
- Position 9 of 10 in Most difficult Calculus problem
7

13
votes
The Collatz Conjecture

This is a simple mathematical problem that has stumped mathematicians for decades. It asks whether a particular sequence of numbers will always eventually reach 1, no matter what starting number is chosen.
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 unsolved mathematical problem
- Position 9 of 10 in Most difficult equation to solve
8

7
votes
The Goldbach Conjecture

Christian Goldbach

This conjecture states that every even number greater than 2 can be expressed as the sum of two prime numbers. It has been tested up to very large numbers but has not been proven.
The Goldbach Conjecture is one of the oldest unsolved problems in number theory. It states that every even integer greater than 2 can be expressed as the sum of two prime numbers.

Year of Conjecture: 1742

Conjecture Type: Unsolved Problem

Field: Number Theory

Difficulty Level: Unknown

Proof Status: Unproven
The Goldbach Conjecture in other rankings
- Position 10 of 10 in Most difficult equation to solve
9

7
votes
The Hodge Conjecture

William Vallancey Duff Hodge

This conjecture connects algebraic geometry to topology and asks whether certain geometric objects can be deformed into simpler objects without losing important information. It is still unsolved.
The Hodge Conjecture is a major unsolved problem in algebraic geometry. It was proposed by Scottish mathematician William Vallancey Duff Hodge in 1950. The conjecture is concerned with the relationship between the topology and algebraic geometry of complex projective manifolds.

Problem type: Unsolved problem

Field: Algebraic geometry

Domain: Mathematics

Difficulty: Very difficult

Conjectured year: 1950
The Hodge Conjecture in other rankings
- Position 6 of 10 in Most difficult math equation
10

3
votes
The Yang-Mills Existence and Mass Gap Problem

This is a problem in theoretical physics that asks whether a certain kind of particle (a massless gauge boson) exists in nature. It is related to the behavior of subatomic particles and has implications for our understanding of the universe.
The Yang-Mills Existence and Mass Gap Problem is a mathematical problem related to quantum field theory, specifically the Yang-Mills theory, which describes the fundamental forces among elementary particles. The problem seeks to prove the existence of Yang-Mills theories with certain properties, particularly the existence of mass for the elementary particles involved. It is considered one of the most challenging and significant open problems in mathematical physics.

Millennium Prize Problem: The Yang-Mills Existence and Mass Gap Problem is one of the seven unsolved Millennium Prize Problems, each of which carries a $1 million prize for a correct solution.

Originated in 1954: Chen Ning Yang first introduced the Yang-Mills Existence and Mass Gap Problem in 1954 during a lecture, and it has since become a prominent question in theoretical physics and mathematics.

Yang-Mills Theory: The problem is intricately connected to the Yang-Mills theory, which describes the behavior of the strong nuclear force, one of the four fundamental forces.

Mass Gap: The problem focuses on proving the existence of a mass gap in Yang-Mills theories, meaning that particles have nonzero mass, which is a crucial aspect of the physical world.

Quantum Field Theory: The Yang-Mills Existence and Mass Gap Problem falls within the realm of quantum field theory, an area of physics that combines quantum mechanics with special relativity to understand particle interactions.

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Ranking factors for difficult math question

Level of mathematical understanding

A difficult math question should require a high level of understanding of mathematical concepts, theorems, and techniques. It should not be easily solvable by someone with a basic grasp of the subject.
Complexity

The question should involve multiple steps or layers, possibly incorporating several different areas of mathematics. The complexity may be in the form of a large number of variables, intricate relationships between elements, or higher-order patterns.
Problem-solving skills

A difficult math question should require creative approaches and a strong grasp of problem-solving techniques. This may include recognizing patterns, applying previously learned concepts in novel ways, or thinking abstractly about relationships between elements.
Time required

The time required to solve a difficult math problem can be an important factor. A challenging question should not be something that can be solved quickly or easily.
Degree of abstraction

A question becomes more difficult if it requires the thinker to handle highly abstract concepts or extend ideas beyond concrete examples.
Ambiguity

Some difficult math questions may have more than one possible solution or may not have a clear answer, requiring the solver to explore alternative approaches or work with uncertainty.
Computational demands

Some math problems require extensive calculations or computations. The sheer number of operations or the complexity of the calculations involved may increase the difficulty of the question.
Intuition

A difficult math question should challenge the solver's intuition and push them beyond their comfort zone.
Conceptual integration

The problem may require integrating concepts from different areas of mathematics, which increases the level of difficulty because the solver must understand and apply multiple concepts simultaneously.
Unfamiliarity

A challenging math question may involve concepts, techniques, or procedures that are not commonly encountered, making it difficult to solve for those who have not experienced them before.

About this ranking

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

Statistics

1905 views
179 votes
10 ranked items

Movers & Shakers

The Riemann Hypothesis
P vs NP
The Four Color Theorem
The Yang-Mills Existence and Mass Gap Problem

Voting Rules

A participant may cast an up or down vote for each Math question once every 24 hours. The rank of each Math question is then calculated from the weighted sum of all up and down votes.