Goldbach's Conjecture
Grade: five stars, diamonds in the crown of mathematics;
Content: Goldbach put forward the following conjecture in his letter 1742 to Euler: any even number greater than 2 can be written as the sum of two prime numbers. But Goldbach himself could not prove it, so he wrote to the famous mathematician Euler to help him prove it, but until his death, Euler could not prove it.
Progress: 1966 Chen Jingrun proved that "1+2" holds, that is, "any sufficiently large even number can be expressed as the sum of two prime numbers, or the sum of a prime number and a semi-prime number". 1956, Wang Yuan proved "3+4"; In the same year, the mathematician A.V. Noguera Dov of the former Soviet Union proved "3+3"; 1957, Wang Yuan proved "2+3"; Pan chengdong proved "1+5" in 1962; 1963, Pan Chengdong, Barba En and Wang Yuan all proved "1+4"; 1966, Chen Jingrun proved "1+2" after making new and important improvements to the screening method.
Riemann hypothesis
Grade: five-star, towering peaks, standing;
Content: All nontrivial zeros of Riemannian function have the real part of 1/2. 1859, Riemann was elected as a member of the Communication Academy of Berlin, and then he submitted a paper entitled "On the number of prime numbers less than a given value" to the Berlin Academy of Sciences. This short eight-page paper is the birthplace of Riemann's conjecture.
Progress: Riemann conjecture 160 The Spring and Autumn Period has passed since its birth. During this period, it was like a towering mountain, which attracted countless mathematicians to climb, but no one could reach the top. According to statistics, there are more than 1000 mathematical propositions based on Riemann conjecture (or its extended form) in today's mathematical literature. If Riemann conjecture is proved, those mathematical propositions can be promoted to theorems; On the other hand, if Riemann's conjecture is falsified, at least some of those mathematical propositions will be buried with him.
Horizon: Fermat's Last Theorem
Grade: five stars, which has puzzled the wise people in the world for 358 years;
Content: 1637, when the French amateur mathematician Fermat was studying the arithmetic of Diophantine, he wrote a few short lines in the book to the effect that no power can be divided by the sum of two powers of the same power except square. I found a good proof, but the margin of this book is too narrow to write down.
Progress: The problem with this prank is the famous Fermat's Last Theorem. This problem has puzzled the mathematical community for 358 years. During this period, Euler, Gauss, Cauchy, Leberg and other famous mathematicians have tried different things, but they all failed. It was not until 1994 that it was solved by the British mathematician andrew wiles.
twin prime conjecture
Grade: five stars, a classic problem in the history of number theory, 17 1 year old "old age";
Content: In 1849, Alfonso de Polignac put forward a general conjecture: for all natural numbers k, there are infinite prime pairs (p, p+2k). The case of k = 1 is the twin prime conjecture. Twin prime numbers refer to prime number pairs with a difference of 2, such as 3 and 5, 5 and 7, 1 1 and 13. This conjecture was formally put forward by Hilbert in the eighth topic of the report of the International Congress of Mathematicians in 1900. It can be described as follows: there are infinitely many prime numbers P, so that p+2 is a prime number. A pair of prime numbers (p, p+2) are called twin prime numbers.
Progress: On April 7th, 20 13, mathematician Zhang contributed to the most prestigious Mathematical Yearbook in the field of mathematics. In Zhang's paper, he gave the result that there are countless pairs of adjacent prime numbers, and the difference between them is only 70 million. But this is only an estimate, and it is not the best result that Zhang's method can get. After the paper was published, some mathematicians thoroughly understood the new method and began to try to improve this constant and further narrow the distance to finally solve the twin prime conjecture. 20/kloc-in February, 2004, the 70 million Zhang Ba has been reduced to 2.46 million.
Poincaré conjecture
Grade: five stars;
Content: 1904, Poincare put forward a seemingly simple topological conjecture in a paper: in a three-dimensional space, if every closed curve can be shrunk to a point, then this space must be a three-dimensional sphere. However, an error was found in 1905, which was revised as: "Any N-dimensional closed manifold that is homotopy with an N-dimensional sphere must be homeomorphic with an N-dimensional sphere." Later, this conjecture was extended to more than three dimensions and was called "high-dimensional Poincare conjecture".
Describe the problem in plain language: the ball in the picture above is covered with a rope, and the two ends of the rope intersect at the yellow point. If you pull the rope to the left and right ends at the yellow spot, you will find that the circle of the noose is gradually narrowing, and finally it can be narrowed to a little point, and the rope can be taken back.
Progress: Poincare conjecture with five or more dimensions was proved by Steven Smale; Four-dimensional Poincare conjecture was proved by michael freedman; Three-dimensional Poincare conjecture was proved by Russian mathematician perelman in 2002-2003. They won the Fields Medal in 1966, 1986 and 2006 respectively. In August 2006, perelman was awarded the Fields Prize, a Nobel Prize in mathematics, in recognition of his contribution to geometry. A medal engraved with Archimedes' embossed head and a prize of about $654.38 +0.35 million were also rejected. In this regard, he gave the reason that "there is no toll to receive the award".
The above are the world-famous mathematical problems I am familiar with, and I look forward to your introduction to other mathematical problems!
(transferred from headline number-mathematical latitude and longitude network)