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

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: http://en.wikipedia.org/wiki/Siegel_zero 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.