Questions tagged [open-problem]

Questions on problems that have yet to be completely solved by current mathematical methods.

There are many unsolved problems in mathematics. Some prominent outstanding unsolved problems (as well as some which are not necessarily so well known) include

  1. The Goldbach conjecture.

  2. The Riemann hypothesis.

  3. The conjecture that there exists a Hadamard matrix for every positive multiple of 4.

  4. The twin prime conjecture (i.e., the conjecture that there are an infinite number of twin primes).

  5. Determination of whether NP-problems are actually P-problems.

  6. The Collatz problem.

  7. Proof that the $196$-algorithm does not terminate when applied to the number $196$.

  8. Proof that $10$ is a solitary number.

  9. Finding a formula for the probability that two elements chosen at random generate the symmetric group $S_n$.

  10. Solving the happy end problem for arbitrary $n$.

  11. Finding an Euler brick whose space diagonal is also an integer.

  12. Proving which numbers can be represented as a sum of three or four (positive or negative) cubic numbers.

  13. Lehmer's Mahler measure problem and Lehmer's totient problem on the existence of composite numbers $n$ such that $\phi(n)|(n-1)$, where $\phi(n)$ is the totient function.

  14. Determining if the Euler-Mascheroni constant is irrational.

  15. Deriving an analytic form for the square site percolation threshold.

  16. Determining if any odd perfect numbers exist.

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Past open problems with sudden and easy-to-understand solutions

What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved thanks to some out-of-the-box flash of genius,…
Damian Reding
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What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif Laczkovich gave a solution with many hundreds of…
Ed Pegg
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Open mathematical questions for which we really, really have no idea what the answer is

There is no shortage of open problems in mathematics. While a formal proof for any of them remains elusive, with the "yes/no" questions among them mathematicians are typically not working in both directions but rather have a pretty clear idea of…
user139000
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What is the oldest open problem in geometry?

Geometry is one of the oldest branches of mathematics, and many famous problems have been proposed and solved in its long history. What I would like to know is: What is the oldest open problem in geometry? Also (soft questions): Why is it so hard?…
Sudoku Polo
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Are there open problems in Linear Algebra?

I'm reading some stuff about algebraic K-theory, which can be regarded as a "generalization" of linear algebra, because we want to use the same tools like in linear algebra in module theory. There are a lot of open problems and conjectures in…
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What are some things we can prove they must exist, but have no idea what they are?

What are some things we can prove they must exist, but have no idea what they are? Examples I can think of: Values of the Busy beaver function: It is a well-defined function, but not computable. It had been long known that there must be…
Loreno Heer
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Primes of the form $n^2+1$ - hard?

I met a student that is trying to prove for fun that there are infinitely many primes of the form $n^2+1$. I tried to tell him it's a hard problem, but I lack references. Is there a paper/book covering the problem? Is this problem really hard or I…
Gadi A
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What are some of the major open problems in category theory?

What are some of the major open problems in category theory? Just curious - I'm interested in category theory.
user122283
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What does proving the Collatz Conjecture entail?

From the get go: i'm not trying to prove the Collatz Conjecture where hundreds of smarter people have failed. I'm just curious. I'm wondering where one would have to start in proving the Collatz Conjecture. That is, based on the nature of the…
Thomas Shields
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Proving the Riemann Hypothesis and Impact on Cryptography

I was talking with a friend last night, and she raised the topic of the Clay Millennium Prize problems. I mentioned that my "favorite" problem is the Riemann Hypothesis; I explained what it posits and mentioned that, if proven, it would have great…
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Status of the classification of non-finitely generated abelian groups.

From the Wikipedia on abelian groups: By contrast, classification of general infinitely-generated abelian groups is far from complete. How far are we from a classification exactly? It seems like divisible groups have been classified. Which cases…
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Can someone explain the ABC conjecture to me?

I am an undergrad and I know that the conjecture may have been proven recently. But in reading about it, I am entirely confused as to what it means and why it is important. I was hoping some of you kind people could help me. I know there are several…
Joseph Skelton
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Is there any palindromic power of $2$?

My question is in the title: Is it possible to find $n≥4$ such that $2^n$ is a palindromic number (in base $10$)? A palindromic number is a number which is the same, independently from which side we read it (forwards and backwards), for example…
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Famous Problems Where We Only Know the Elementary

Define a graph with vertex set $\mathbb{R}^2$ and connect two vertices if they are unit distance apart. The famous Hadwiger-Nelson problem is to determine the chromatic number $\chi$ of this graph. For the problem as stated (there are many…
Austin Mohr
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Has anyone found a "pattern" in prime numbers?

Yesterday I was having some fun trying to look for some patterns in primes; and I think I found something interesting. The idea is to start with an array of primes {p1, p2, p3, ... }, print it, then set the value at index i = abs( [i] - [i-1] ) or…
KDecker
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