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10 Math Equations That Have Never Been Solved

By Kathleen Knowles, 10 Sep 2020

Mathematics has played a major role in so many life-altering inventions and theories. But there are still some math equations that have managed to elude even the greatest minds, like Einstein and Hawkins. Other equations, however, are simply too large to compute. So for whatever reason, these puzzling problems have never been solved. But what are they?

Like the rest of us, you're probably expecting some next-level difficulty in these mathematical problems. Surprisingly, that is not the case. Some of these equations are even based on elementary school concepts and are easily understandable - just unsolvable.

1. The Riemann Hypothesis

Equation: σ (n) ≤ Hn +ln (Hn)eHn

  • Where n is a positive integer
  • Hn is the n-th harmonic number
  • σ(n) is the sum of the positive integers divisible by n

For an instance, if n = 4 then σ(4)=1+2+4=7 and H4 = 1+1/2+1/3+1/4. Solve this equation to either prove or disprove the following inequality n≥1? Does it hold for all n≥1?

This problem is referred to as Lagarias’s Elementary Version of the Riemann Hypothesis and has a price of a million dollars offered by the Clay Mathematics Foundation for its solution.

2. The Collatz Conjecture

Equation: 3n+1

  • where n is a positive integer n/2
  • where n is a non-negative integer

Prove the answer end by cycling through 1,4,2,1,4,2,1,… if n is a positive integer. This is a repetitive process and you will repeat it with the new value of n you get. If your first n = 1 then your subsequent answers will be 1, 4, 2, 1, 4, 2, 1, 4… infinitely. And if n = 5 the answers will be 5,16,8,4,2,1 the rest will be another loop of the values 1, 4, and 2.

This equation was formed in 1937 by a man named Lothar Collatz which is why it is referred to as the Collatz Conjecture.

3. The Erdős-Strauss Conjecture

Equation: 4/n=1/a+1/b+1/c

  • where n≥2
  • a, b and c are positive integers.

This equation aims to see if we can prove that for if n is greater than or equal to 2, then one can write 4*n as a sum of three positive unit fractions.

This equation was formed in 1948 by two men named Paul Erdős and Ernst Strauss which is why it is referred to as the Erdős-Strauss Conjecture.

4. Equation Four

Equation: Use 2(2∧127)-1 – 1 to prove or disprove if it’s a prime number or not?

Looks pretty straight forward, does it? Here is a little context on the problem.

Let’s take a prime number 2. Now, 22 – 1 = 3 which is also a prime number. 25 – 1 = 31 which is also a prime number and so is 27−1=127. 2127 −1=170141183460469231731687303715884105727 is also prime.

5. Goldbach's Conjecture

Equation: Prove that x + y = n

  • where x and y are any two primes
  • n is ≥ 4

This problem, as relatively simple as it sounds has never been solved. Solving this problem will earn you a free million dollars. This equation was first proposed by Goldbach hence the name Goldbach's Conjecture.

If you are still unsure then pick any even number like 6, it can also be expressed as 1 + 5, which is two primes. The same goes for 10 and 26.

6. Equation Six

Equation: Prove that (K)n = JK1N(q)JO1N(q)

This equation tries to portray the relationship between quantum invariants of knots and the hyperbolic geometry of knot complements. Although this equation is in mathematics, you have to be a physics familiar to grasp the concept.

7. The Whitehead Conjecture

Equation: G = (S | R)

  • when CW complex K (S | R) is aspherical
  • if π2 (K (S | R)) = 0

What you are doing in this equation is prove the claim made by Mr. Whitehead in 1941 in an algebraic topology that every subcomplex of an aspherical CW complex that is connected and in two dimensions is also spherical. This was named after the man, Whitehead conjecture.

8. Equation Eight

Equation: (EQ4)

This equation is the definition of morphism and is referred to as an assembly map.  Check out the reduced C*-algebra for more insight into the concept surrounding this equation.

9. The Euler-Mascheroni Constant

Equation: y=limn→∞(∑m=1n1m−log(n))

Find out if y is rational or irrational in the equation above. To fully understand this problem you need to take another look at rational numbers and their concepts.  The character y is what is known as the Euler-Mascheroni constant and it has a value of 0.5772.

This equation has been calculated up to almost half of a trillion digits and yet no one has been able to tell if it is a rational number or not.

10. Equation Ten

Equation: π + e

Find the sum and determine if it is algebraic or transcendental. To understand this question you need to have an idea of algebraic real numbers and how they operate. The number pi or π originated in the 17th century and it is transcendental along with e. but what about their sum? So Far this has never been solved.


As you can see in the equations above, there are several seemingly simple mathematical equations and theories that have never been put to rest. Decades are passing while these problems remain unsolved. If you're looking for a brain teaser, finding the solutions to these problems will give you a run for your money.

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