9 Hardest Calculus Problems Ever You’ll Ever Encounter

Hardest Calculus Problems Ever
Hardest Calculus Problems Ever

Lately, I was teaching one of the brightest students; she asked me what the hardest calculus problem ever was. Her question led me to do deeper research to find.

Mathematics is a constantly evolving field, and new equations and calculations are constantly being discovered. But some problems have posed a challenge for mathematicians for centuries. Here are the

In this blog, I wanted to share what I found; maybe you try solving it. Read on to find some of the most challenging calculus problems and discuss why they are so difficult.

The good news is two of the hardest calculus problems are still unsolved, and there is a reward of $1 million dollars for whoever finds the answers to each problem.

You might also enjoy reading: What Jobs Can You Get With a Mathematics Degree: 9 Best Options.

1- The Three-Body Problem

The Three-Body Problem is one of the oldest and most famous unsolved problems in mathematics. It was first proposed by Isaac Newton in 1687 and remains unsolved to this day (Source: Scientific American)

The three-body problem deals with understanding the motion of three objects interacting with each other, such as moons orbiting planets or stars in galaxies, given their initial positions and velocities.

It has been particularly difficult for mathematicians due to its chaotic behavior, meaning that small changes in the initial conditions can lead to drastically different outcomes. Additionally, its nonlinearity makes it resistant to traditional mathematical techniques.

When Isaac Newton published his Principia in 1687, he asked: “How will two masses move in space if the only force on them is their mutual gravitational attraction?” Newton formulated the question as a problem solving a system of differential equations

Despite these challenges, many researchers have made significant progress on the Three-Body Problem over the years, but it still remains unsolved (Source: Popular Mechanics)

Watch the video below to learn more about the Three-Body Problem.

2- Goldbach’s Conjecture

Christian Goldbach first proposed this conjecture in 1742 and stated that every even number greater than two could be written as the sum of two prime numbers (a prime number is an integer greater than one with no divisors other than itself).

For example, 8 = 3 + 5 or 10 = 7 + 3. While this conjecture seems simple enough at first glance, it has proven surprisingly hard to prove or disprove!

Despite intense effort from mathematicians worldwide over 250 years, Goldbach’s Conjecture remains unproven and stands as one of the greatest open problems in mathematics today.

Further progress on Goldbach’s conjecture emerged in 1973 when the Chinese mathematician Chen Jing Run demonstrated that every sufficiently large even number is the sum of a prime and a number with at most two prime factors.

Hardest Calculus Problem Ever
Goldbach partitions of the even integers from 4 to 50 rev4b” by Adam Cunningham and John Ringland is licensed under CC BY-SA 3.0.

3- Fermat’s Last Theorem

This theorem dates back to 1637 when Pierre de Fermat wrote down his famous equation without providing any proof or explanation for it in his notebook: “it is impossible to separate a cube into two cubes or a fourth power into two fourth powers or generally any power higher than second into two like powers.”

Fermat’s last theorem, also known as Fermat’s great theorem, is the statement that there exist no natural numbers (1, 2, 3,…) x, y, and z such that x^n + y^n = z^n, in which n is a natural number bigger than 2.

 For instance, if n = 3, Fermat’s last theorem says that no natural numbers xy, and z exist such that x^3 + y ^3 = z^3. In other words, the sum of two cubes is not a cube (Source: Britannica)

The Fermat’s Last Theorem remained unproven until 1995 when Andrew Wiles finally provided proof using elliptic curves after working on it for seven years (Source: National Science Foundation (NSF))

This theorem stands as one of the greatest achievements in mathematics and still remains one of the most difficult problems ever tackled by mathematicians worldwide.

4- The Riemann Hypothesis

The Riemann Hypothesis is perhaps one of the most famous unsolved problems in mathematics today. It states that all non-trivial zeros of the Riemann zeta function have real parts equal to 1/2.

While it has not yet been proven (or disproven), mathematicians have made considerable progress towards solving it using techniques from complex analysis and number theory.

Unfortunately, many mathematicians believe it may never be solved without major mathematics and computer science breakthroughs due to its complexity and difficulty.

If you are looking for ways to make a million dollars by solving math, try solving the Riemann Hypothesis. It is among the Seven Millennium Prize Problems, with a $1 million reward if you find its solutions. 

If you solve the Riemann Hypothesis tomorrow, it will open an avalanche of further progress. It would be massive news throughout the topics of Number Theory and Analysis. 

I suggest you watch the video below to learn more about the Riemann Hypothesis.

5- The Collatz Conjecture

The Collatz Conjecture is another unsolved mathematical problem that has remained a mystery since its inception in 1937. Intuitively described, it deals with the sequence created by taking any number and, if it is even, dividing it by two, and if it is odd, multiplying by three and adding one.

Every cycle of this algorithm eventually converges to the same number: 1. So far, no one has been able to determine why this happens or why the Collatz Conjecture holds true for all natural numbers (positive integers from 1 to infinity).

This elusive problem has stumped mathematicians for decades and continues to draw researchers to try and solve this head-scratching conundrum.

Despite numerous attempts made to unravel its secrets, the Collatz Conjecture remains as enigmatic as ever, begging us to discover its mystery and open up new doors in the realm of mathematics.

If proved true, the Collatz Conjecture could provide major new insights into our understanding of mathematics and computing algorithms, leading to numerous potential applications.

Undoubtedly, whoever solves this hypothesis will have made one of the great discoveries in mathematics.

The video below discusses the Collatz Conjecture.

6- The Twin Prime Conjecture

In number theory, the Twin Prime conjecture, also known as Polignac’s conjecture, asserts that infinitely many twin primes, or pairs of primes, differ by 2. As an illustration, 3 and 5, 5 and 7, 11 and 13, and 17 and 19 are considered twin primes. As numbers become larger, primes become less frequent, and twin primes are rarer still.

The Twin Prime Conjecture is an unsolved problem in mathematics that has stumped the best minds for centuries. If the conjecture is true, it will open up a whole new realm of prime numbers, providing new avenues for exploration and even potential applications in cryptography.

However, it has been difficult to prove due to the lack of general patterns for consecutive primes; any pattern made thus far is inconsistent and unreliable at best.

Despite this difficulty, mathematicians remain optimistic about uncovering the answer to this mystery–and when they do, it will surely be a monumental achievement!

I encourage you to watch the video below to learn more about the Twin Prime Conjecture.

7- The Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer Conjecture, a crucial unsolved mathematical problem in number theory, remains one of the greatest mysteries of our time. The Birch and Swinnerton-Dyer Conjecture is also among the six unsolved Millennium Prize Problems, meaning that if you solve it, you will be rewarded with one million dollars.

Originally conjectured in the 1960s, this idiosyncratic conjecture has captivated mathematicians ever since. While researchers have gained insight into related topics such as elliptic curves and modular forms, the true complexity of this conjecture has still eluded them.

An elliptic curve is a particular kind of function that can be written in this form y²=x³+ax+b. It turns out that these types of functions have specific properties that explore other math topics, such as Algebra and Number Theory.

As a result, researchers continue to explore new approaches and hope they can one day demonstrate their veracity. Undoubtedly, this intriguing yet tough problem will captivate mathematicians for years to come.

If you are interested in learning more about the Birch-Swinnerton-Dyer Conjecture, I encourage you to watch the video below.

8- The Kissing Number Problem

The Kissing Number Problem has stumped mathematicians for centuries. The problem involves finding the maximum number of equal-sized spheres that can touch one central sphere without overlapping or leaving any spaces between them (Source: Princeton University)

Initially thought to be a simple problem to solve, it is quite challenging to determine this ‘kissing number accurately.’ The answer varies depending on the dimension of the space – in two dimensions, or a flat surface, it’s only six, but in three dimensions, it is much larger and still debated today.

The Kissing Number Problem continues to baffle modern mathematicians, providing an interesting and complex challenge that could lead to countless scientific advances.

Watch the video below to learn more about the Kissing Number Problem.

9- The Unknotting Problem

The Unknotting Problem has fascinated mathematicians since discovering that the unknot is equivalent to a one-dimensional closed loop in three-dimensional space.

The Unknotting Problem simply asks if a particular knot can be undone without changing its form. For example, questions such as “which knots have the fewest crossings?” or “can all knots be unknotted?”. It was first described in 1904 by Greek mathematician Peter Guthrie Tait. It is still an open problem with no known general algorithm for efficiently deciding whether a knot can be untied to become just a circle.

As fascinating as this perplexing problem is to mathematicians, understanding and solving the Unknotting Problem could be essential for researchers hoping to apply mathematics to biology and chemistry, where twists and turns play an important role in the workings of molecules.

Check out A Journey From Elementary to Advanced Mathematics: The Unknotting Problem if you want to learn more.

Here is an interesting PPT presentation about the Unknotting Problem.

Hardest Calculus Problem Ever
Jkasd, Public domain, via Wikimedia Commons

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Wrapping Up

Mathematicians have been tackling difficult mathematical problems since time immemorial with varying degrees of success; some have been solved, while others remain unsolved mysteries today.

From Goldbach’s Conjecture to Fermat’s Last Theorem, challenging mathematical problems continue to captivate mathematicians everywhere and provide them with fascinating puzzles to solve.

Whether you are a high school student studying calculus or a college student taking a higher-level math course, there is always something interesting waiting for you if you look for it.

I encourage you to give some of these hardest math problems a try. Who knows – maybe you will come up with an answer and be rewarded with a million dollars prize.


I am Altiné. I am the guy behind mathodics.com. When I am not teaching math, you can find me reading, running, biking, or doing anything that allows me to enjoy nature's beauty. I hope you find what you are looking for while visiting mathodics.com.

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