Something interesting about the article (and how analytic number theory is done in general).
As mentioned in the article, it was proved over 50 years ago that every odd N can be written as a sum of three primes, provided N is sufficiently large. The proof extends to cover five primes instead, if you like (indeed, write N = 2 + 2 + (N - 4), but in fact the five primes case is in fact easier).
Usually in such proofs the definition of "sufficiently large" is maybe like 10^(10^(10^(10^(10^10000)))) or something similarly absurd, possibly far worse, and it is typically difficult even to calculate such a fixed N, because then you can't use big-O estimates in your calculations. Sometimes you can't even compute such an N, it is "ineffective", see e.g. here for an example:
As mentioned in the article, it was proved over 50 years ago that every odd N can be written as a sum of three primes, provided N is sufficiently large. The proof extends to cover five primes instead, if you like (indeed, write N = 2 + 2 + (N - 4), but in fact the five primes case is in fact easier).
Usually in such proofs the definition of "sufficiently large" is maybe like 10^(10^(10^(10^(10^10000)))) or something similarly absurd, possibly far worse, and it is typically difficult even to calculate such a fixed N, because then you can't use big-O estimates in your calculations. Sometimes you can't even compute such an N, it is "ineffective", see e.g. here for an example:
http://en.wikipedia.org/wiki/Siegel_zero
Tao's accomplishment, which was really cool, was to bring the value of N down to the level where you could settle the small N case by brute force.