What is the hardest math problem ever to solve?

Question

Is there a certain one that is longest, most difficult, most time consuming, etc?

If so, I would like to see.

Answer 1

Fermat apparently made a note saying the proof was trivial, but, never wrote it down before he died. It was insanely difficult to solve, but, Fermat had said that it was provable. Mathematicians transformed the problem into another domain and solved it there then showed how the proof would also apply to the original domain. Standard operating procedure in mathematics. Took a while to find and recognize the appropriate parallel domain in which to prove the original.

Back in Pythagoras's day, his followers were a secretive band and most mathematical knowledge was passed on verbally only to followers who had proved themselves loyal. They had an idea that all numbers were rational. Then one day a follower of Pythagoras was passing some time on a boat trip and decided to work out the diagonal of a 1 x 1 square. We know now that this is root(2) = 1.414213562..... an irrational number. I.E. no (integer/integer) will exactly equal root(2).

This was a difficult problem. The difficulty arose because the belief system of the Pythagoreans at the time was such that irrational numbers do not exist and that Pythagoras was perfect, etc.... I don't know if this guy was ejected from the "cult" or if he killed himself or if maybe others suicided over the scandal, but, I think this would be a good candidate for a "difficult" problem.

As for a long-standing time consuming problem, Fermat's theorem proof is the most famous.

Finding all the digits of pi is not possible, but, trying to list out billions and billions of them is a time consuming task but is not difficult. You need to find an efficient algorithm and a fast computer but then you can sit back and let the computer generate the digits.

What we need is a problem that is long-standing like the Fermat Theorem proof, but is difficult in that it's solution and perhaps even the problem statement are in contention, like the Pythagorean accidental discovery of irrational numbers.

How about this one then:

Problem statement: Our current cryptology algorithms result in absolutely unbreakable security. Prove that this is correct.

See, this problem has the attributes of being a very long-standing, time-consuming problem because we have been using ciphers for centuries and someone always finds a way to defeat them -brute force, man-in-the-middle attacks and so on. Now with computers it is easy to put together complex algorithms for encryption that may seem to be unbreakable but there is always the worry that they can be broken.

The social difficulty arises with this problem because on one hand we don't want you to find that our encryption schemes are like swiss cheese but on the other hand we do want more secure schemes..

Also I have phrased the question to prove the current schemes are good which is more difficult a proof than to simply show by a counterexample that they can be broken -sort of like the old saying "can't prove a negative"

whew. that was fun. Now you have a problem that has been around a long time. Is most difficult (technically and socially) and is very likely going to be very time consuming to prove.

Answer 2

Hardest Math Problem Ever Solved

Answer 3

There have been and there are many very difficult math problems. Fermat's theorem is just one of the many. David Hilbert, one of the foremost mathematicians of all times, proposed a set of 23 very hard problems at the turn of the 20th century. Only a couple are open still. One of them is the Riemann hypothesis, having to do with properties of the prime numbers. The Clay Institute (see web reference below) defined in 2000 a number of very hard mathematical problems (the "new" Hilbert problems) and opened them to the mathematical community for consideration as part of the community's research programme. Check the website links I have attached below. Interesting material...

Answer 4

If you mean one that was very hard to solve but WAS solved, Fermat's Last Theorem is a good candidate. Another is Poincare's Conjecture, solved recently by a Russian mathematician.

There are lots of unsolved problems, however. The most famous one has to be Riemann's Hypothesis. Solve that one, and fame and fortune are yours.

Answer 5

This Site Might Help You.

RE:
What is the hardest math problem ever to solve?
Is there a certain one that is longest, most difficult, most time consuming, etc?

If so, I would like to see.

Answer 6

Fermat's Last Theorem?

It is this:
"If an integer n is greater than 2, then the equation a^n + b^n = c^n has no solutions in non-zero integers a, b, and c."

It was stated in 1637, and eventually proven in 1995. There were a lot of pretty smart people working on it, for over 300 years.

Answer 7

Fermat's Last Theorem and there are still open problems in mathematics

Answer 8

Fermats last theorem

Answer 9

to try and find pi which you would have to be 100% accurate and then see if it will ever end. It may end but they have already figured 1,000,000,000,000 digits out but have fun lol.