There are about half a dozen problems that almost all mathematicians agree are supremely important. One that I particularly like is the 鈥淧 = NP鈥� problem.
This concerns a situation that occurs not just in mathematics but also in everyday life: often it is easy to recognise a solution to a problem, but not at all easy to find one. For example, if you are asked to factorise 10541, you will have to spend a long time searching for factors amongst the primes. But if you are told that 10541 is 83 脳 127, it is routine to check that it is true. A non-mathematical example is finding anagrams of words.
So far, so uncontroversial. However, one of the great gaps in our knowledge is that nobody knows how to show that searching for solutions really is harder than checking that the solutions are correct. This is the P = NP problem.
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This problem gets to the heart of mathematics, because mathematical research itself has the property I have described: it seems to be easier to check that a proof is correct than to discover it in the first place. Therefore, if we found a solution to the P = NP problem it would profoundly affect our understanding of mathematics, and would rank alongside the famous undecidability results of Kurt G枚del and Alan Turing.