In this episode there were several mathematical topics that were mentioned. We'll discuss the Riemann hypothesis, other Millenium problems, and encryption.
As was made obvious in the episode, the Riemann Hypothesis is one of the most famous conjectures in mathematics. It was originally stated in an 1859 paper written by Bernhard Riemann, and it involves the Riemann zeta-function defined as the infinite series:
It can be shown that this series converges if the real part of s (which is a complex number in general) is greater than 1. Then it is possible to analytically continue the zeta function so that it is defined for all s with real part not equal to 1. This definition of the zeta function is analytic on its domain of definition, which is the complex analysis version of being differentiable. This basically means that with this formula we can define the function for part of the complex plane, and then we can show that there is a way of extending this to almost all of the complex plane that agrees nicely with the original definition. The Riemann Hyopthesis is that all the zeros of the continued zeta function occur either when s is a negative even integer or when the real part of s is 1/2. It is important in mathematics because it has many deep connections to prime numbers and especially to the distribution of the prime numbers (how often they occur), and there are many interesting results which have been proved to be true assuming that the Riemann hypothesis is true. For more information about various connections between other parts of math, look up the Wikipedia page about it.
As was also mentioned in the episode, the Riemann hypothesis is one of the Millennium Prize Problems. These are 7 (actually, the number recently became 6) famous mathematical problems which come with a prize of one million dollars offered by the Clay Mathematics Institute for anyone who solves them. This institute was established by Landon T. Clay and is "dedicated to increasing and disseminating mathematical knowledge." Here is a list of the problems with short descriptions:
One of the main plot points of the episode is that the bad guys wanted the math professor to give them his solution to the Riemann hypothesis so that they could crack the encryption on major financial data. As was mentioned in the episode, many encryption algorithms depend on the fact (or apparent fact) that it is very hard to factor large numbers. One of the main encryption algorithms used today is called the RSA algorithm (the name comes from the initials of its inventors. This algorithm is also described Episode 205. As mentioned there, the main difficulty in trying to break the RSA code is that factoring numbers into prime factors can be very difficult. In particular, there is no known algorithm that factors a number n into its prime factors that is in P, i.e. runs in polynomial time as a function of log(n) (here log is used because the number of bits it takes to store a number n is log base 2 of n). It is believed that no such algorithm exists. Interestingly, there is another kind of cryptosystem that is provably difficult in the sense that if an algorithm is devised to decrypt messages in polynomial time, then this algorithm can be adapted to factor integers in polynomial time.