Pure mathematicians generally do not like secrecy. It would be just too childish to keep such a word to oneself. They also know well to not assume the result as proved until a refereed journal confirms it. All that would have happened if the word got out to NYT would be a PR mess about which the mathematical community couldn't care less.
Isn't it generalizing too much? Andrew Wiles worked on Fermat's last theorem for 7 yrs in secret, nobody even knew he was working on it until he published the results.
Wiles is widely regarded as odd, to put it mildly, for that effort. And it still didn't work and he had to get a former student in to help fix the proof.
That seems true in general, but it's still dangerous to declare that you're working on a famous problem (P!=NP, 3n+1, Riemann); it's something you keep to yourself until you've made serious progress.