The Millennium Prize Problems are seven of the most well-known and important unsolved problems in mathematics. The Clay Mathematics Institute, a private nonprofit foundation devoted to mathematical research, famously challenged the mathematical community in 2000 to solve these seven problems, and established a US $1,000,000 reward for the solvers of each. One of the seven problems has been solved, and the other six are the subject of a great deal of current research. Let’s take a look.

The only millennium problem that has been solved to date is Poincare Conjecture. This problem was posed in 1904 about the topology of objects called manifolds. The most basic problem in topology is to determine when two topological spaces are the same that is they can be identified with one another in a continuous way. The conjecture is essentially the first conjecture ever made in topology, it asserts that a three dimensional manifolds is the same as the 3D sphere precisely when a certain algebraic condition is satisfied. The conjecture was formulated by Poincare around the turn of 20 Poincare Conjecture: thcentury.

Riemann Hypothesis: The Riemann Hypothesis is perhaps the most famous unsolved problems in mathematics. It concerns the non-trivial zeros of the Riemann Zeta Function. The generalized Riemann Hypothesis is a statement about the zeros of certain functions known as L functions defined by Barclayseries which are the generalization on Riemann Zeta Functions.

Birch-Swinnertoner conjecture: In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. The conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute, which has offered a $1,000,000 prize for the first correct proof. It is named after mathematicians Bryan Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. As of 2021, only special cases of the conjecture have been proven. Mordell (1922) proved Mordell’s theorem: the group of rational points on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite subset of the rational points on the curve, from which all further rational points may be generated. If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. The number of independent basis points with infinite order is called the rank of the curve, and is an important invariant property of an elliptic curve. If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points. On the other hand, if the rank of the curve is greater than 0, then the curve has an infinite number of rational points. Although Mordell’s theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve. The rank of certain elliptic curves can be calculated using numerical methods but (in the current state of knowledge) it is unknown if these methods handle all curves. An L-function L(E, s) can be defined for an elliptic curve E by constructing an Euler product from the number of points on the curve modulo each prime p. This L-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form. It is a special case of a Hasse-Weil L-function.

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