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I am not a math's guy, however I like maths that have to do with puzzle, not just solving an exercise.

I was wondering which is the simplest yet difficult math question ever been in the form of an equtation ?

Thank you and excuse me with this not so much math related question.

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This is a pretty subjective question, so may not be very well suited to this site. However, I'd suggest that Goldbach's conjecture (that every even integer greater than 2 is the sum of two primes) is a strong candidate for 'simplest difficult problem'. It's not normally expressed as an equation, though you could do that if you wanted to. –  Chris Taylor Jun 17 '11 at 16:50
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This is not a good type of question for this site, because it is purely a matter of opinion what constitutes a dificult question (and the difficulty of a question often depends on what the person trying to answer it knows or has seen before). Please read the faq and look around the site for the kinds of questions that the site is designed to address, or try to edit your question to make it more constructive. –  Arturo Magidin Jun 17 '11 at 16:54
    
perhaps a rephrasing of the question, and making it community wiki would help: Something to the effect: "What math question(s) which first appeared rather simple to you, turned out to be much more complicated than ever would have imagined"...So it's not a global assessment of "simple" or "difficult"...There are lots of "list" type questions...I would encourage JBSalut to rephrase the question, before voting to close? –  amWhy Jun 17 '11 at 17:02
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Hmmm...Chandru, if this was in need of closing, then why the h*** did you post an answer?! –  amWhy Jun 17 '11 at 17:17
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@amWhy: Look, I am not interested in arguing. I voted to close because, the question was subjective, I posted an answer because, I felt it would be benefit the OP. I don't want to say anything else. Whatever you want to comment do so.. –  user9413 Jun 17 '11 at 17:32
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closed as not constructive by Arturo Magidin, yunone, Chandru, Ross Millikan, Qiaochu Yuan Jun 17 '11 at 17:14

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3 Answers

I think Fermat-Wiles theorem is the simplest equation-like math problem with the most difficult known answer (in proportion):

Are there any three nonzero integers $x,y,z$ satisfying $x^n+y^n=z^n$ for any natural power $n>2$?

No, there are not!

The amount of incredibly interesting but advanced math developed over centuries and needed to prove this overwhelms the simplicity of its statement. This is very amusing since there are infinite Phythagorean triples which are solutions to the problem for exponent $n=2$. The general formula for them is interestingly related to very simple geometric constructions and serves as an example of how algebraic geometry has a connection to arithmetic problems. The same is true for Fermat's theorem whose proof by contradiction is based on showing that a particular "curve" built up from the formula cannot exist.

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How do you know that there is no easy proof for the theorem? :-) –  Listing Jun 17 '11 at 17:11
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Since there has not been found any easy proof for the theorem, at the present time its proof is extremely difficult in proportion to the simplicity of the stament. The fact that after so long we do not know easier proofs does not mean there aren't any but this fact says a lot about the huge difficulty of the problem itself qualifying the solution to be considered very difficult to get, even if it eventually ends up being simple. Indeed it is subjective, maybe a very advanced extraterrestial civilization considers as elementary the current proof by Wiles :-) –  Javier Álvarez Jun 17 '11 at 18:08
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I would go for the Collatz conjecture

Take any natural number n. If n is even, divide it by 2 to get n / 2, if n is odd multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1.

Even little kids can understand that and it is unsolved since 60 years (although a "new proof" came up recently it seems it is flawed. So the conjecture seems save for now...)

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Well there are many such:

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Hi there thank you for this. Goldbach conjecure is a theorem right? We need to prove that this is correct? –  JBSalut Jun 17 '11 at 17:13
    
How is anything related to the E-M constant simple?? I need to go back to school ... :) –  The Chaz 2.0 Jun 17 '11 at 17:14
    
@JBSalut: it still remains unproven. Much work has been done to verify that the conjecture holds up to very very large values, but that falls short of "proving for all integers"... –  amWhy Jun 17 '11 at 17:15
    
yes it is still a conjecture –  user9413 Jun 17 '11 at 17:16
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@chaz: the statement of the question of E-M is simple –  user9413 Jun 17 '11 at 17:19
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