The most difficult equation in mathematics

[Dragonfall]

Yes, if N=1 :D

Or P = 0

 

[jk22]

How About finding the configuration of charges minimal that is stable in electrostatics (i just invented that one)

 

[ChrisVer]

Well I am not sure what you mean by most difficult equation in mathematics. Any differential equations that needs computers to solve (have no analytical solution) seems "difficult". An example are the coupled Boltzmann equations for the Big Bang Nucleosynthesis (it's fresh in my mind).

 

[ChrisVer]

Yes, if N=1 :D

It can as well be N=P if P is idempotent. Eg the projection operators when written in matrix representation satisfy that P^2 =P

 

[Demystifier]

1+1=2

It's not so difficult. Whitehead and Russell needed only 379 pages to prove this.
http://en.wikipedia.org/wiki/Principia_Mathematica
http://quod.lib.umich.edu/cgi/t/tex...3201.0001.001&frm=frameset&view=image&seq=401

 

[jk22]

You can also build a semi group where 1+1=1 it is just a matter of symbols we use to describe the operations.

 

[micromass]

It's not so difficult. Whitehead and Russell needed only 379 pages to prove this.
http://en.wikipedia.org/wiki/Principia_Mathematica
http://quod.lib.umich.edu/cgi/t/tex...3201.0001.001&frm=frameset&view=image&seq=401

Ah, but take a look at your post here:

The problem has been considered a lot. The most important result (by Cohen) is a proof that the problem is unsolvable by using standard axioms of set theory. Different non-standard axioms of set theory may lead to different solutions, but then the problem is how to know which axioms, if any, are the "right" ones?

For more details see
http://en.wikipedia.org/wiki/Continuum_hypothesis

Just like $2^{\aleph_0} = \aleph_1$ depends on the axiomatics used, so does 1+1=2

 

[micromass]

Also, whether 1+1=2 is difficult depends a lot on your definitions of 1 and 2. It is certainly possible to find a definition where 1+1=2 is trivially true by definition. For example, in the Peano system, 1+1=2 can be very easily verified. The hard part is when 1 and 2 are taken as real or complex numbers. In that case, the definitions are much more complicated and do not allow for a trivial solution. And then there are other number systems such as the hyperreals or surreals which make things even more complicated! So the difficulty lies more in the number systems used than in the actual equation.

 

[micromass]

You should still note that no single algorithm exists for finding the integral of a function, even if we exclude integrals which can't be expressed in terms of elementary functions(Oops...its not applicable to such integrals, right?). So finding such an algorithm is an open problem and it seems to be very hard.

http://en.wikipedia.org/wiki/Risch_algorithm

 

[lavinia]

What, in your opinion, seems to be (one of) the most difficult equation(s) in mathematics?

Here is my choice:
Find the solution $k$ of the equation
$$2^{\aleph_0}=\aleph_k$$

What do you mean by difficult?

How about solving the Riemann hypothesis? This remains unsolved. The equation is Riemann zeta function = 0.

Solving the Laplace equation has inspired generations of research.
Solving the Heat equation has also led to generations of research.
The same holds for other PDE's e.g. Monge-Ampere equations, geometric evolution equations, the wave equation ...

The proof of Fermat's last theorem led to centuries of research. The solution shows that certain equations have no solution.

 

[phoenixthoth]

Here's one that I think is hard:

math = ?

 

[micromass]

math = math

 

[dextercioby]

Not necessarily equation, but counting problems are always terribly hard. Here's one that's really difficult: How many DISTINCT valid (within the rules of the game) endings of a standard 9x9 Sudoku game are there?