Let's Start The Week,

with not one but two brain teasers to solve,

in previous weeks the problems have been very easy, but here are two that need a lot of thought, but they are not without their rewards, including world recognition for solving them, the first above, the Riemann Hypothesis, some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern, however, the German mathematician G.F.B. Riemann (1826 - 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function  ζ(s) = 1 + 1/2s + 1/3s + 1/4s + ...called the Riemann Zeta function, the Riemann hypothesis asserts that all interesting solutions of the equation, ζ(s) = 0 lie on a certain vertical straight line, this has been checked for the first 10,000,000,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers, this problem is unsolved,

and here is a second, the Birch and Swinnerton-Dyer Conjecture,

mathematicians have always been fascinated by the problem of describing all solutions in whole numbers x,y,z to algebraic equations like  x2 + y2 = z2 Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. Indeed, in 1970 Yu. V. Matiyasevich showed that Hilbert's tenth problem is unsolvable, i.e., there is no general method for determining when such equations have a solution in whole numbers. But in special cases one can hope to say something. When the solutions are the points of an abelian variety, the Birch and Swinnerton-Dyer conjecture asserts that the size of the group of rational points is related to the behavior of an associated zeta function ζ(s) near the point s=1. In particular this amazing conjecture asserts that if ζ(1) is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if ζ(1) is not equal to 0, then there is only a finite number of such points, this problem is again unsolved, but here is the thing if you solve either of these problems you will receive $1 million for each one, even better news, if you look here you will find these problems and several others, each netting $1 million per problem, so a mug of coffee or two and you could become a millionaire, several times over!