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Goldbach Conjecture: Closer To Solved?
mikejuk writes "The Goldbach conjecture is not the sort of thing that relates to practical applications, but they used to say the same thing about electricity. The Goldbach conjecture is reasonably well known: every integer can be expressed as the sum of two primes. Very easy to state, but it seems very difficult to prove. Terence Tao, a Fields medalist, has published a paper that proves that every odd number greater than 1 is the sum of at most five primes. This may not sound like much of an advance, but notice that there is no stipulation for the integer to be greater than some bound. This is a complete proof of a slightly lesser conjecture, and might point the way to getting the number of primes needed down from at most five to at most 2. Notice that no computers were involved in the proof — this is classical mathematical proof involving logical deductions rather than exhaustive search."
Terry Tao always amazes me with the scope of his knowledge. Contributions in mathematical areas as diverse as random matrix theory, harmonic analysis, and number theory. I look forward to what comes next!
Notice that no computers where involved in the proof — this is classical mathematical proof involving logical deductions rather than exhaustive search.
Exhaustive search for a result that holds for every integer? Good luck with that one.
That's not completely true. Wiles's proof only proves it for an exponent that is a prime p>=7. So one needs the classical results of n=3,4,5,7 also. This is to some extent a minor criticism. Your essential point is correct that sometimes a proof of a theorem comes out of a completely different direction. But, very often, it does come from a straightforward way of refining the same techniques more and more. For example, Catalan's Conjecture http://en.wikipedia.org/wiki/Catalan's_conjecture (the claim that that the only consecutive positive perfect powers are 8 and 9) was proven by what in many ways amounted to slow and steady progress.