Mathematics has many open conjectures which are ridiculously hard to even understand. But this is not always the case. An example is:

  • Collatz conjecture.

I would like to see some more examples. So my question is:

What are some easy to explain conjectures which are very hard to prove and unsolved?

Like most big list questions, please include one conjecture per answer. Most of the big list questions have many answers, so, for the sake of convenience of the reader who wants to read all the answers, I am including the answers:

  • Goldbach's conjecture. It states that every even integer greater than 2 can be expressed as a sum of two primes.
  • Twin prime conjecture: There are infinitely many twin primes.
  • $\begingroup$ Would it be good if I edit the question and add the conjectures given in the answers to make it convenient for the reader to find all the conjectures given in the answers? I will put "the conjectures given in the answers are:" before writing them. $\endgroup$ – Euler 2 Nov 11 '20 at 8:24
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    $\begingroup$ There may be a lot of overlap with mathoverflow.net/questions/75698/… and mathoverflow.net/questions/100265/… $\endgroup$ – Gerry Myerson Nov 11 '20 at 8:41
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    $\begingroup$ Also, your statement about $n^x$ is misleading. What is open is this: if $2^x$ and $3^x$ are integers, then $x$ is an integer. $2$ and $3$ can be replaced, if you're careful, with $m$ and $n$. $\endgroup$ – Gerry Myerson Nov 11 '20 at 8:44
  • $\begingroup$ @GerryMyerson yeah, but many answers are problems, like the moving sofa problem. I am asking about conjectures, which ask whether a statement is true or false. So I think this question shouldn't be closed. But yeah, there is a lot of overlapping. $\endgroup$ – Euler 2 Nov 11 '20 at 8:46
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    $\begingroup$ You have edited your statement about $n^x$, but you've still got it wrong. It was a Putnam problem, to show that if $n^x$ is an integer for every (positive) integer $n$, then $x$ is an integer, so it's pretty far from being open. More is true: it's known that if $2^x,3^x,5^x$ are all integers, then $x$ is an integer (and again with some care one can replace $2,3,5$ with $r,s,t$). $\endgroup$ – Gerry Myerson Nov 11 '20 at 12:02

Goldbach conjecture: Every even number greater than 2 is the sum of two primes.


Twin prime conjecture: There are infinitely many twin primes. Twin primes are pairs of primes with difference 2.


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