Top 5 Hardest Calculus Problems In The World

Airplane, representing Navier-Stokes Equations and how they relate to flight. The Navier-Stokes Equations are among the hardest calculus problems in the world.

Have you ever wondered what the hardest Calculus problems in the world are?

I’m currently midway through my Calculus 2 course and already on the verge of mental exhaustion. Yet, I desperately want to know what the most difficult Calculus problems are and what it might take to solve them. Also, I want to know what real-world applications they might have if they were solved.

I did some research and what I found out was really exciting. There are actually several unsolved Calculus problems which, if solved, could have some revolutionary real-world applications across several fields.

Additionally, two of the problems that made this list could earn a person $1,000,000, awarded by the Clay Mathematics Institute if a solution is found. These 5 unsolved problems are among the hardest in the world that fall into the realm of Calculus.

Top 5 Hardest Calculus Problems

1. Navier-Stokes Existence And Smoothness Equation

2. Riemann Hypothesis

3. Euler Equations (Fluid Dynamics)

4. Vlasov Equations

5. Inversion Formula for Broken Polar Ray Transform

These are 5 of the hardest unsolved Calculus problems. Arguably, some of these problems fall beyond the realm of Calculus, being that they are Partial Differential Equations (PDE). However, I’m including them because they are remarkable problems and they’re at least rooted in Calculus.

5. Inversion Formula for Polar Broken Ray Transform

The Polar Broken Ray transform was introduced in 2015 by Brian Sherson in his 140-page Doctorate thesis on the subject that can be found here:

Brian Sherson: Some Results In Single-Scattering Tomography

Brian Sherson’s work was built on the work of Lucia Florescu, John C. Schotland, and Vadim A. Markel in their 2009 study of the Broken Ray transform.

An inversion formula was found in 2014 for the 2009 study of the Broken Ray transform. However, no inversion formula has been found for Brian Sherson’s 2015 Polar Broken Ray transform.

Difficulty Rating


While being extremely difficult to solve, this currently unsolved Calculus problem is far from impossible. If research continues, we should have a solution within the next decade. While the solution wouldn’t earn it’s discoverer a $1,000,000 prize, it would likely earn them an honorary doctorate in mathematics.

Real-World Applications

These equations describe the scattering of photons as they’re transferred through an object. Therefore, they have tremendous applications in advanced X-rays, also known as CT scans (Computed Tomography).

4. Vlasov Equation

The Vlasov equation was first considered in 1938 to describe plasma.

The equation was developed by Anatoly Vlasov. However, it was further used in conjunction with James Clerk Maxwell’s equation and Simeon Denis Poisson’s equation. This resulted in the Vlasov-Maxwell system of equations, as well as the Vlasov-Poisson equation.

Difficulty Rating


This has been unsolved for over 80 years and will likely remain unsolved for some time.

In order to solve this problem, one would need a deep understanding of physics and mathematics including distribution functions.

Real-World Applications

The Vlasov-Maxwell system of equations describes the interactions of plasma particles.

The Vlasov-Poisson equations estimate the Vlasov-Maxwell equations.

3. Euler Equations (Fluid Dynamics)

The Euler equations, named after Leonhard Euler, were originally presented in 1755 and later published in 1757. These equations are closely linked with another of the most difficult Calculus problems, the Navier-Stokes equations, which comes in at #1 on this list.

The Euler equations, when concerned with fluid dynamics, are a set of quasilinear hyperbolic equations which govern the adiabatic and inviscid flow. Of these equations, there is a general form of the continuity equation, the momentum equation, and the energy balance equation.

Leonhard Euler also created one of the most famous equations among the mathematics community, simply called Euler’s Identity which linked 5 important mathematical constants: 0, 1, i, e, and pi.

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Regardless of the fact that these equations are over 250 years old, they remain mostly unsolved. For instance, in three space dimensions, it’s still unclear if solutions are defined for all time or if they’re singularities.

Real-World Applications

The Euler equations have applications in thermodynamics, hydrodynamics, and aerodynamics.

2. Riemann Hypothesis

The Riemann hypothesis was originally hypothesized in 1859 by Bernhard Riemann. It is one of the seven Millenial Prize Problems that will earn it’s decoder $1,000,000 paid by the Clay Mathematics Institute. It’s one of two Millenial Prize Problems that made this list.

The Riemann hypothesis speculates that the Riemann zeta function crosses the x-axis (the functions zero’s) only at negative even integers and complex numbers with real part 1/2.

This hypothesis is thought to be the most important unsolved problems in mathematics, let alone in Calculus.

Difficulty Rating


Not only is this one of the most important problems in Calculus, but it’s also one of the hardest, if not the hardest, by far. There has been some progress through the years to validate the Riemann hypothesis, but a formal proof has yet to be given.

Real-World Applications

The Riemann hypothesis has extensive applications in number theory, the branch of mathematics dealing with whole numbers, especially prime numbers.

1. Navier-Stokes Existence and Smoothness Equation

Similar to the Euler equations that come in at #3 on this list, the Navier-Stokes existence and smoothness equation is at the heart of fluid dynamics. This means that they describe how fluids, along with fluid-like substances such as air, move through space.

The Navier-Stokes existence and smoothness equation was developed in 1822 by Claude-Louis Navier and George Gabriel Stokes. This equation appears on the Clay Mathematics Institute’s list of Millenial Prize Problems that will pay out $1,000,000 to whoever solves it.

The official statement they would have proved or disproved is as follows:

In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.

Clay Mathematics Institute

Difficulty Rating


This is one of the most important problems in physics that has plagued the mathematics and scientific community since its inception. In order to solve this problem, one would need a deep understanding of advanced Calculus, but more specifically, differential equations. However, some have speculated that a solution would be physically impossible.

Real-World Applications

The Navier-Stokes existence and smoothness equation has applications in fluid mechanics, aerodynamics, and in the engineering of aircraft. A proper solution would be revolutionary in the aerospace industry.

If you’d like to learn about the other Millenial Prize Problems, you can check them out here:

Wikipedia: Millenial Prize Problems

More Unsolved Math Problems

If you’re interested to learn more about the hardest unsolved math problems in the world, Wikipedia has a list of 100+ problems. The list is organized by the different branches of mathematics such as algebra and number theory. You can check out the list here:

Wikipedia: List of unsolved problems in mathematics

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Tim Statler

Tim Statler is a Computer Science student at Governors State University and the creator of Comp Sci Central. He lives in Crete, IL with his wife, Stefanie, and their cats, Beyoncé and Monte. When he's not studying or writing for Comp Sci Central, he's probably just hanging out or making some delicious food.

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