What are some hypotheses in mathematics that cannot be made or are not meaningful?

The Riemann conjecture, which can be said to be one of the most important conjectures in mathematics, is the study of the distribution of prime numbers, and prime numbers are the basis of all numbers, if human beings master the law of the distribution of prime numbers, then can easily solve many well-known mathematical problems. However, the difficulty of the Riemann conjecture can be said to be unprecedented, and even some mathematicians desperately believe that human beings may never be able to grasp the law of prime distribution, and the Riemann conjecture itself is unprovable.

In the world of mathematics and computer science, there are many problems that we know how to solve quickly with computer programs, such as basic arithmetic, sorting problems, data searching, and so on. These problems can be solved in "polynomial time" (p). It means that the steps required to complete tasks such as adding and sorting a list are affected at the polynomial level, such as the number of numbers, the length of the list, and so on.

For example, if the running time of a program increases by the same amount as the size of the data increases, then we call the time complexity of the program o(n). For example, the algorithm for finding the maximum value in n numbers. The program needs to iterate through all the values to get the maximum value.

As the size n of the input data increases, so does the traversal time.

In general, the mathematical rule of algebraic geometry is the study of high-dimensional shapes that can be defined algebraically as a set of solutions for algebraic equations. To take the simplest example, if you remember y=x2 in algebra in middle school, when the solution of the equation is drawn on a piece of paper, you get the shape of a parabola. Algebraic geometry deals with a higher latitude version of a curve when considering a system of complex equations with multiple multinomials.

In the 20th century, mathematicians developed many more sophisticated techniques to better understand the objects of algebraic geometry, such as curves, surfaces, and hyperboloids. These unimaginable shapes can be made more accessible by complex calculation tools. The Hodge conjecture suggests that certain geometries have a particularly useful algebraic correspondence for studying and classifying these shapes.

One of the oldest and most extensive objects of mathematical research is the Diophantine equation, or polynomial equation for which we want to seek integer solutions. One of the most classic examples is the number of Pythagorean triples, or three sets of integers that satisfy Pythagorean theorem, that is, the Pythagorean theorem x2+y2=z2, that we learned in junior high school geometry class. Elliptic curves have been studied for more than 200 years.

An elliptic curve is a curve defined by a special class of Diophantine equations. These curves have important applications in both number theory and cryptography, and finding integer or rational solutions to these curves is the main research in this field.

The ns equation is a set of differential equations. Differential equations are used to describe how a particular quantity changes over time under given initial conditions, and they can be used to describe almost all physical systems. In the example of the ns-equation, starting from the initial flow of the fluid, we can use differential equations to describe the evolution of the flow of this fluid over time.

But the N.S. equation is much more difficult: mathematically, the techniques currently used to solve other differential equations are not effective against it; From a physical point of view, fluids can exhibit chaotic and turbulent behaviors – for example, the initial flow of smoke from candles and cigarettes tends to be smooth and can be **, but will soon fall into an unavoidable vortex.

The Riemann hypothesis, which limits these existences by establishing ranges based on the distance of the distribution of prime numbers from the mean, is a conjecture about a mathematically constructed zero-point distribution called the "Riemannian function." The Riemann function is a special curve in the plane of complex numbers, and the function has become an independent research topic in the field of mathematics, which makes the Riemann hypothesis and the problems related to it all the more important.

Mathematics and physics have always been in a mutually beneficial relationship. The development of mathematics often opens up new avenues for the study of physical theories, and new physical discoveries often provoke deeper fundamental mathematical explanations. Quantum mechanics is one of the most successful physical theories of all time.

Matter and energy behave very differently at the atomic and subatomic scales, and one of the greatest achievements of the 20th century was the development of a set of theories and experiments to understand this behavior.

Is there an analytic solution to the Navier-Stokes equation? This equation describes the problem of viscous fluid flow, which is a partial differential equation with extremely complex solutions, and can only be solved numerically within a certain range.