I know statements for some unsolved problems related in some way to number theory, being stated (I refer for each one of these unsolved problems) several equivalent formulations, for example: these are the Riemann hypothesis, Goldbach conjecture (see the equivalent statement from [1]), the twin prime problem in terms of particular values of some number theoretic functions (see as reference [2]), if I refer well the determination of even perfect numbers (in terms of particular values of the sum of divisors function, and also Lucas-Lehmer test determining what integers are Mersenne primes)..., or as a last example to get a closed-form for the Apèry's constant computing integrals or series as Dirichlet series (see [3]).
All these problems are related to the category of number theory (or analytic number theory).
Question. I would like to know* remarkable unsolved problems, in different categories of mathematics than number theory/analytic number theory, for which were stated several equivalent formulations. Many thanks.
I refer that I know the previous problems thus I am not interested precisely in the mentioned in the introductory paragraph.
*Only it is required a reference in which was published the equivalent formulation, and if you want as is known the unsolved problem or a general reference for this unsolved problem that you refer in the mathematical bibliography.
I've flagged the post because I feel that the downvote that was casted is harassement.
References:
[1] The article dedicated to Goldbach Conjecture from the encyclopedia Wolfram MathWorld.
[2] The article dedicated to Twin Primes from the encyclopedia Wolfram MathWorld.
[3] Tom M. Apostol, Introduction to analytic number theory, UTM Springer (1976).
User @barrycarter suggested $P=NP$. Let's expand on that.
Loosely speaking, $P$ is the set of problems that can be solved in time polynomial in the length of the input; $NP$ is the set of problems for which a proposed solution can be checked in time polynomial in the length of the input. Clearly, $P\subseteq NP$. The question is whether $P=NP$
A problem is said to be NP-complete if every problem in $NP$ can be reduced to it in polynomial time. If an NP-complete problem is in $P$, then $P=NP$, and conversely, if $P=NP$, then every NP-complete problem is in $P$.
So we see that for every NP-complete problem $X$, the statement "$X$ is in $P$" is equivalent to $P=NP$.
Now, there are hundreds – maybe thousands, by now – of problems that have been proved to be NP-complete. See. for example, https://en.wikipedia.org/wiki/List_of_NP-complete_problems where $130$ NP-complete problems are listed.