Introduction Finding primes Counting primes Modeling primes Finding patterns in primes Structure and randomness in the prime numbers Terence Tao University of California, Los Angeles July 19 / IMO Terence Tao Structure and randomness in the primes Introduction Finding primes Counting primes Modeling primes Finding patterns in primes The Distribution Of The Primes The prime numbers 2,3,5,7, . . . are one of the oldest topics studied in mathematics. We now have a lot of intuition as to how the primes should behave, and a great deal of confidence in our conjectures about the primes... ... but we still have a great deal of difficulty in proving many of these conjectures! Ultimately, this is because the primes are believed to not obey behave pseudorandomly in many ways, and not to follow any simple pattern. We have many ways of establishing that a pattern exists... but how does one demonstrate the absence of a pattern? Terence Tao Structure and randomness in the
