The most difficult equation in mathematics

So what about finding k for any given m in [itex] 2^{\aleph_m}=\aleph_k[/itex]?
Seems a nice challenge! Is it possible to give a more difficult equation than a given one?
Anyway, I think no such thing(most difficult equation) exists in mathematics. Or at least its existence isn't a trivial fact!

 

I'm wondering which is harder! Finding k for a given m or finding m for a given k?
Has such a problem been solved in any of its cases?(or maybe in general!!!)(I was going to write 'considered' instead of 'solved' but I thought it surely is considered by someone! or not?)

 

Difficult and solved by traditional methods might be Fermat's Last solved by Andrew Wiles. Difficult and unsolved might be nonsense. How is difficulty measured, modus ponens ponendo?

 

Demystifier used "most difficult equation"! Those are proofs of facts. Of course proving things can become arbitrarily difficult. (EDIT: This is not trivial too!)
And about measuring difficulty. I don't think we need to get rigorous here. If a good mathematician says that something is difficult, Its enough for me to believe that thing is difficult. (I hope you don't ask for a definition of a good mathematician!!!)

 

I think you should use "desired" instead of "right"!
Anyway, why just use 2? We can generalize to [itex] n^{\aleph_m}=\aleph_k [/itex]. Right? Or there is some reason for that 2?

P.S.
Mathematical logic and set theory, or anything which seems that fundamental is really interesting for me but there lots of things to learn, and things with applications to physics have priority. A day will come that I'll learn those!:D

 

Looks like I wasn't clear enough. Actually I meant Is it that we can only place 2 as the base, because of some features of aleph numbers, or its possible to put any other number as the base? Does that base have to be a natural number? An integer? Real?

 

I saw that book before. It was thick enough to make me think it contains much physics but I really didn't expect it to contain such a pure math subject! Thanks.

 

The continuum hypothesis says that k = 1. The continuum hypothesis is undecidable (Gödel theorem).

 

Solving any of the equations in this list in the most general case!(Sorry for ruining the chance for others but I wanna emphasize my last sentence in post#2!)

 

These problems were and are also quite difficult to solve:
http://en.wikipedia.org/wiki/Hilbert's_problems
http://www.claymath.org/millennium-problems

A difficult problem that can be stated in simple terms: construct an algorithm that will find, if it exists, the general analytic solution of a first order ordinary differential equation y'=f(x), with f a specified (analytic) function.

Anyway, I think we can only sum up some unsolved problems in this thread. It is difficult to say 'this one is more difficult than that one, because it requires blabla theory' without a proper definition of what makes a problem 'difficult'.

 

This is a bit outside of my area of expertise, but I think this one could be added to the list. http://en.wikipedia.org/wiki/P_versus_NP_problem

Edit: Not sure if this really counts as an equation...