Terence Tao

"Understand the problem. What kind of problem is it? There are three main types of problems: ‘Show that ...’ or ‘Evaluate ...’ questions, in which a certain statement has to be proved true, or a certain expression has to be worked out; ‘Find a...’ or ‘Find all...’ questions, which requires one to find something (or everything) that satisfies certain requirements; ‘Is there a ...’ questions, which either require you to prove a statement or provide a counterexample (and thus is one of the previous two types of problem)."

"Relying on intelligence alone to pull things off at the last minute may work for a while, but generally speaking at the graduate level or higher it doesn't. One needs to do a serious amount of reading and writing, and not just thinking, in order to get anywhere serious in mathematics."

"[P]rofessional mathematics is not a sport (in sharp contrast to mathematics competitions). The objective in mathematics is not to obtain the highest ranking, the highest score, or the highest number of prizes and awards; instead, it is to increase understanding of mathematics (both for yourself, and for your colleagues and students), and to contribute to its development and applications."

"For the third law.., Kepler had... 6 data points..: every planet.., the length of the orbit and the distance to the sun.., and he did... regression. ...He could fit a curve to these 6 data points and he got a square cube law. Amazing, but he was... lucky. That's not enough data to be... reliable."

"There was a later astronomer, Johann Bode who took the same data... and... had a prediction that the distances to the planets formed a shifted ... He also fit a curve, except there was one point missing, a... gap... His law predicted... a missing planet. ...[W]hen Uranus was discovered by Herschel, the distance fit... this pattern. Then Ceres was discovered... in the astroid belt, and it also fit the pattern. ...But then Neptune was discovered, and it was... way off. ...[I]t was a numerical fluke."

"Gauss... created one of the first mathematical data sets. He computed the first [~]100,000 prime numbers. ...He found a statistical pattern ...they get sparser and sparser, but the drop-off in the density was inversely proportional to the of the range of numbers. So he conjectured... the the number of primes up to x is... x divided by the natural log of x, and he had no way to prove this. It was a conjecture. It was revolutionary because it was maybe the first important conjecture of math that was statistical in nature. ...It just gave you an approximation that got better... as you went further... out. ...It started the field of... . ...[I]t was the first of many ...which ...started consolidating the idea that the prime numbers ...really didn't have a pattern, that they behaved like random sets of numbers with a certain density. They have some patterns. They're almost all odd. ...They're not actually random. They're... pseudorandom."

"Over time it became more... productive to think of the primes as if they were generated by some... random set, and this allowed us to make... other predictions. ...[B]ecause of this statistical random model.., we are ...absolutely convinced it's true. ...[W]e have, over time, developed this very accurate conceptual model of what primes should behave like.., but it's mostly and non-rigorous, but extremely accurate. It's the same reason we believe the Riemann hypothesis is true, and why we believe that cryptography based on the primes is... mathematically secure."

"[O]ne reason why we care about the Riemann hypothesis is that if... we knew it was false, ...it would mean there was a secret pattern to the primes that we were not aware of, and I think we would very rapidly abandon any cryptography based on the primes, because if there was one pattern... there are probably more, and these... can lead to exploits in crypto."