20 12 In August, Shinichi Mochizuki, a mathematician from Kyoto University in Japan, claimed to have proved this conjecture. However, because no one can understand its research tools and papers, it is impossible to verify whether it is correct. This conjecture has not been solved so far.
Introduction to Conjecture abc Conjecture was first proposed by Joseph Austler and David massl in 1985. It shows that for any ε >; 0, with a constant c ε > 0, for any three positive integers A, B, C that satisfy a+ b= c and A, B are coprime, there are:
Where rad(n) represents the product of prime factors of n, such as rad(72) = rad (2×2×2×3×3) = 2×3 = 6.
In 1996, Ellen Baker put forward a more accurate conjecture, using rad(n).
Substitution, where ω is the number of different prime factors of a, b, c, b and c.
Abc conjecture includes many Diophantine problems, such as Fermat's last theorem. Like many Diophantine problems, abc conjecture is completely a problem of prime relations. Brian Conrad of Stanford University once said, "There is a deeper correlation between the prime factors of A, B and a+b".
Project content ABC@home is a mathematical project operated by the Institute of Mathematics of Leiden University in the Netherlands based on BOINC distributed computing platform. It aims to obtain the distribution of these arrays by searching for ternary arrays satisfying ABC conjecture conditions, thus helping mathematicians solve this conjecture. |
That is, it exhaustively uses decentralized computing until C.
By studying the distribution of these ternary arrays, this project tries to find a way to prove ABC conjecture, a mathematical unsolved problem. If ABC conjecture is proved, Fermat-Cartland conjecture can be partially proved, and Zinzel-Thiedemann conjecture can be completely proved. The concrete content of ABC conjecture is that for all e>0, there is a constant C(e) related to E, a, b and c < = c (e) ((rad (ABC)) (1+e)). There are many evidences to support ABC conjecture, such as the polynomial version of ABC conjecture, which also contains Fermat's last theorem. D. Goldfeld evaluated ABC conjecture as "the most important unsolved problem in the field of Diophantine analysis (that is, the analysis of integral coefficient equations and solutions)". ABC@home hopes to help mathematicians solve ABC conjecture by knowing the distribution of ternary arrays that meet the conditions.
Research Progress Many mathematicians have spent a lot of energy trying to prove this conjecture. In 2007, based on the research work of French mathematician Lucien Szpiro 1978, the proof of abc conjecture was published for the first time, but it was soon found that the proof was flawed.
In 2006, the Department of Mathematics of Leiden University in the Netherlands and the Kennislink Institute of Science in the Netherlands jointly launched a BOINC project called "ABC@Home" to study this conjecture.
20 12 August, Shinichi Mochizuki, a mathematician from Kyoto University, published a 500-page proof of abc conjecture. Although the whole proof process has not been proved to be correct, some famous mathematicians, including Tao Zhexuan, have given positive comments on it.
Japanese mathematician Shinichi Mochizuki's research significance comes from Columbia University mathematician Dorian Goldfeld's comment: "abc conjecture, if proved, will solve many famous Diophantine problems in one fell swoop, including Fermat's last theorem. If the proof of full moon is correct, it will be one of the most shocking mathematical achievements in 2 1 century. "
Comparison between the definition in the full moon paper and the traditional concept in number theory: the research work of the new moon has little to do with the previous efforts. He established a brand-new mathematical method and used some brand-new mathematical "objects"-these abstract entities can be compared to familiar geometric objects, sets, permutations, topologies and matrices, which only a few mathematicians can fully understand. As Goldfield said, "At present, he may be the only one who has completely mastered this method."
Conrad believes that this research work "contains many profound ideas, and it will take a long time for the mathematics community to fully understand digestion." The whole proof contains four long papers, each of which is based on the previous paper. "It takes a lot of time to learn and understand these profound long proofs, so we can't just pay attention to the importance of this proof, but more importantly, we should study it along the author's proof ideas."
Shinichi Mochizuki's research results make all these efforts worthwhile. Conrad said: "Shinichi Mochizuki has successfully proved extremely difficult theorems, and his paper is rigorous and fully discussed. All these make us full of confidence in successfully proving abc conjecture. " In addition, he added that the achievement is not limited to the confirmation of this certificate. "The exciting reason is not only that abc conjecture may have been solved, but also that the methods and ideas he uses will become a powerful tool to solve the problem of number theory in the future."
Countless counter-intuitive theories have been proved to be correct in history. Once the counterintuitive theory is proved to be correct, it basically changes the process of scientific development. For example: Newton's law of inertia, if an object is not subjected to external force, it will maintain its current state of motion, which is undoubtedly a heavyweight ideological bomb in the17th century. "When an object is not stressed, it will of course change from motion to stop", which was the normal idea of ordinary people based on daily experience at that time. In fact, this idea would be naive for anyone who studied junior high school physics in the 20th century and knew that there was a force called friction. But for people at that time, the inertia theorem was indeed quite contrary to human common sense!
ABC conjecture is just like Newton's law of inertia in17th century, which is contrary to common sense of mathematics for ordinary people and researchers of number theory. This common sense is: "The qualitative factors of A and B should have nothing to do with the qualitative factors of their sum." One of the reasons is that allowing addition and multiplication to interact algebraically will lead to infinite possibilities and unsolvable problems, such as Hilbert's tenth question about the unified methodology of Diophantine equations, which has long been proved impossible. If ABC conjecture is proved to be correct, there must be a mysterious relationship between addition, multiplication and prime numbers that has never been touched by known mathematical theories.