5 tasks for the solution of which will give a million dollars

Mathematics, as you know, "the queen of the sciences." Those who are engaged in it seriously - special people - they live in the world of formulas and numbers

. The knowledge of the world of mathematics, there are practical sense: for addressing a number of problems the Clay Institute is ready to give a million dollars

. 1. The Riemann Hypothesis

We all remember since high school series of numbers that can be divided only by itself and one. They are called simple (1, 2, 3, 5, 7, 11, 13, 17 ...). The largest known prime numbers today was found in August 2008 and consists of 12,978,189 digits.




For mathematicians, these numbers are very important, but how they are distributed according to the number series is still not completely clear. In 1859, the German mathematician Bernhard Riemann proposed a method of searching and testing, finding a method by which we can determine the maximum number of primes not exceeding a certain predetermined number. Mathematics was subjected to inspection, this method has on a half trillion of prime numbers, but no one can prove that further verification is successful.

This is not a simple "mind games." The Riemann Hypothesis is widely used in the calculation of security of data transmission, so the proof is of great practical significance.

2. Navier-Stokes equations

Navier-Stokes equations are the basis for calculations in geophysical fluid dynamics, including the flows of movement in the Earth's mantle. Use these equations and in aerodynamics. Their essence is that any movement is accompanied by changes in the environment, twists and flows.

For example, if the boat is sailing on the lake, then diverge from its motion waves of aircraft generated turbulence.

These processes, if to simplify and describe started as early as the first third of the XIX century, the Navier-Stokes equations. The equations are, but to solve them still can not. Moreover, it is unknown whether the solutions they exist.

Mathematicians, physicists and engineers have successfully used these equations, substituting them are already known values ​​of velocity, pressure, density, time and so on. If someone will use these equations in the opposite direction, that is, calculating the equation parameters, or to prove that the method is not the solution, then this "someone" will become a dollar millionaire.

3. Hodge Conjecture

In 1941, the Cambridge professor William Hodge suggested that any geometric body can be studied as an algebraic equation and make it a mathematical model. If you come from the other side to the description of this hypothesis, we can say that it is more convenient to explore any object if it can be decomposed into its component parts, and these parts have to investigate.

But here we are faced with the problem of investigating the separately taken a stone, we can not say anything about the actual fortress, which is built of such stones, how many rooms in it, and how they form. Furthermore, in the preparation of the initial components of the object (which we disassembled) superfluous parts can be detected, or vice versa - short of

. Hodge Achieving that he described the conditions under which will not occur "extra" parts, and will not need to get lost. And all this with the help of algebraic calculations. Neither prove his hypothesis, nor deny mathematics may not have 70 years. If this happens you - become a millionaire

. 4. Hypothesis Birch and Svinerton-Dyer

Equations of the form xn + yn + zn + ... = tn have been known to mathematicians of antiquity. The decision of the simplest of them (the "Egyptian triangle" - 32 + 42 = 52) has been known in Babylon. Its fully explored in the III century BC mathematician Diophantus of Alexandria, in the fields of "Arithmetic" which Pierre Fermat formulated his famous theorem.

In the pre-computer era of the greatest solution of this equation it was proposed in 1769 by Leonhard Euler (2682 + 4404 + 15 365 6394 18 796 7604 20 615 = 6734). Common, universal method of calculation for such equations there, but we know that each of them can be either finite or infinite number of solutions.

In 1960, mathematicians Birch and Svinerton-Dyer, experiment on a computer with some famous curves, managed to create a method, which reduces each such equation to a simpler, called the zeta function. According to their hypothesis, if this function at the point 1 is equal to 0, then the number of solutions will be sought endless. Mathematicians have assumed that the property will be maintained for all curves, but neither prove nor disprove this assumption is no one could. In which the assumption of mathematicians will not work to get the coveted million, need to find an example.



5. The problem of the Cook-Levin

The problem-solving test the Cook-Levin is that any decision on the verification takes less time than to solve the problem itself.

If you clearly: we know that somewhere in the bottom of the ocean is a treasure, but do not know where it is. His search can take place so indefinitely. If we know that the treasure is in such a square, a certain specified coordinates, the treasure hunt considerably simplified. Always like this. Likely. So far, none of the mathematicians, and mere mortals could not find such a problem, the solution of which would take less time than checking the correctness of its decision. If suddenly you will find this - immediately write to the Clay Institute. If the Commission approves the mathematicians - a million dollars in your pocket

. The problem of the Cook-Levin was formulated back in 1971, but until now no one is solved. Its solution can be a real revolution in cryptography and encryption systems because there will be "perfect codes", which will be hacking virtually impossible.

Author: Alex Rudevich