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## Numb3rs 106: Prime Suspect | ||||||||

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As was also mentioned in the episode, the Riemann hypothesis is one of the
Millennium Prize Problems. These are 7 (actually, the number recently
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< < | dollars offered by the [[http://www.claymath.org/][ Clay Mathematics Institute]] for anyone who solves them. | |||||||

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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: |

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## Numb3rs 106: Prime Suspect
## Riemann Hypothesis and Millenium ProblemsAs 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
As was also mentioned in the episode, the Riemann hypothesis is one of the
- P versus NP: This problem deals with how hard different problems are to solve on a computer based on the length of the input for the problem. (Here the particular problem is fixed and the length of its input is allowed to vary.) A problem S is in the set P if there is a polynomial
*p(x)*and a computer algorithm which given an input for S of length*n*can solve the problem in time*p(n)*. An example is the problem of sorting a list of*n*numbers, which can be done in polynomial time. A problem is an NP problem if given an input and a possible solution, the solution can be checked to see if is an actual solution in polynomial time (again the polynomial is a function of the length of the input). An example of this kind of problem is the zero sum subset problem - given a set of integers is there a subset of them that sum to zero. Clearly if a problem is in P it is also in NP. The major question is whether there is a problem that is in NP but is not in P. - The Hodge Conjecture: This conjecture is even difficult to state since it involves some very technical definitions. A (very) vague statement is that certain algebraic invariants of algebraic/geometric objects called varieties come from geometric invarients.
- The Riemann Hypothesis: This was discussed above.
- Yang-Mills existence: This is discussed in Episode 103.
- Navier-Stokes existence and smoothness: The Navier-Stokes equations describe in full detail the motion of liquids. Even though they have been known for over a century, it is still not known whether there are smooth (i.e. differentiable infinitely many times) solutions to these equations.
- The Birch and Swinnerton-Dyer conjecture: This conjecture is similar to the Hodge conjecture in that it is difficult to state, but it roughly states that there is a relationship between the number of solutions to a particular type of equation and the order of the zero of the L-function at a particular point (L-functions are related to Riemann's zeta function.)
- The Poincare conjecture: This conjecture has actually been solved in 2006 and recieved fairly wide media coverage. The two-dimensional version of the the conjecture says that if you have a surface that is simply connected (that means that if you draw a loop on the surface, you can shrink it to a point by sliding the loop continuously along the surface) and closed (it doesn't go off to infinity, and can't be stretched so that it does), then it can be deformed into a sphere. The three dimensional version of this took many years to prove.
## Encryption
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 |

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