Riemann hypothesis
Riemann conjecture is one of the most famous problems in mathematics, which was put forward by German mathematician Riemann in19th century. Riemann conjecture involves the distribution law of prime numbers in complex number fields. Specifically, Riemann conjecture points out that all nontrivial zeros lie on a straight line, which is called "the critical line of Riemann conjecture".
The importance of Riemann conjecture is that it involves the distribution law of prime numbers. Prime number is a basic concept in mathematics, which is widely used in cryptography, computer science and other fields. If Riemann conjecture is proved, then we will be able to better understand the distribution law of prime numbers and apply it to a wider range of fields.
Horizon: Fermat's Last Theorem
Fermat's last theorem is one of the most famous problems in mathematics, which was put forward by the French mathematician Fermat in the17th century. Fermat's last theorem points out that for any natural number n greater than 2, the equation x n+y n = z n has no positive integer solution.
The proof of Fermat's Last Theorem lasted nearly 400 years. It was not until 1994 that the British mathematician andrew wiles gave a complete proof. This proof is one of the most famous proofs in the history of mathematics, which involves a lot of profound mathematical knowledge.
Poincaré conjecture
Poincare conjecture is one of the most important problems in mathematics, which was put forward by French mathematician Poincare in the 20th century. Poincare conjecture involves three-dimensional spherical problems in topology. Specifically, it points out that any three-dimensional compact manifold is an isomorphic equivalent class of three-dimensional sphere.
The proof of Poincare conjecture lasted nearly a hundred years, and it was not until 2003 that Russian mathematician grigory perelman gave a complete proof. This proof is one of the most famous proofs in the history of mathematics, which involves a lot of advanced mathematics knowledge, such as manifold, Riemannian metric, calculus and so on.
Godel's incomplete theorem
Godel's incomplete theorem is one of the most important problems in mathematics, which was put forward by Austrian mathematician Godel in the 20th century. Godel's incomplete theorem involves the axiomatic system in mathematical logic. Specifically, it points out that there are some unprovable propositions in any axiomatic system containing natural numbers.
The importance of Godel's incomplete theorem lies in that it reveals the limitations of mathematics. It tells us that mathematics is not complete, and it has some unprovable propositions. This is of great significance for us to understand the essence of mathematics.