1. Arithmetic research mainly refers to Gauss's representative arithmetic research 180 1 year, and Gauss's representative arithmetic research came out. Arithmetic research is written in Latin. This book was written on the eve of graduation from Gauss University, and it took three years before and after. 1800, Gauss sent the manuscript to the French Academy of Sciences, requesting publication, but it was rejected, so Gauss had to publish it with his own funds. The content, scope, academic significance, core topics, congruence theory, quadratic reciprocity law and complex number play an important role in the development of theoretical number theory. The core topics, academic significance, congruence theory, quadratic reciprocity law, theoretical number theory development and complex number play an important role in the understanding of complex number first. In this book, complex number is included in the category of number theory. At the beginning of the preface, Gauss clearly explained the scope of this book: "This book studies the integer part of mathematics, excluding fractions and irrational numbers." [Edit this paragraph] Arithmetic research in academic significance is an epoch-making work, which ended the unsystematic state of number theory before19th century. In this book, Gauss systematically sorted out all outstanding and sporadic achievements of predecessors in number theory, actively promoted them, gave standardized marks, classified research problems and known solutions, and introduced new methods. [Edit this paragraph] The core theme book * * * has three core themes: congruence theory, homogeneity theory and residue theory, and quadratic reciprocity law. These are all outstanding contributions made by Gauss to number theory. Congruence theory congruence is a basic research topic in arithmetic research. This concept was not first put forward by Gauss, but it was Gauss who introduced modern symbols into congruence and studied it systematically. He discussed in detail the operation of congruence, the basic theorem of polynomial congruence and the congruence of power. He also proved Fermat's last theorem by using the congruence theory of power. Quadratic reciprocity law Quadratic reciprocity law is one of the proudest achievements of Gauss and occupies an extremely important position in number theory. As Dixon (1874- 1954), a modern American mathematician, said, "it is the most important tool in number theory and occupies a central position in the development history of number theory." In fact, Gauss got this theorem and its proof as early as 1796. What was published in Arithmetic Research is another proof. The development of quadratic reciprocity law begins with quadratic reciprocity law. Gauss has successively derived biquadratic reciprocity law and cubic reciprocity law, and the related biquadratic and cubic residue theories. In order to make cubic and biquadratic residue theory beautiful and simple, Gauss developed complex integer and complex integer number theory. And its further achievement must be algebraic number theory, which is the decisive contribution made by Dai Dejin (1831-1916), a student of Gauss. [Edit this paragraph] In type theory's arithmetic research, Gauss unusually discussed type theory, the longest one. After abstracting the concept of type equivalence from Lagrange's works, he put forward a series of type equivalence theorems and type compound theory with great enthusiasm. His work effectively shows the importance of types-used to prove theorems about any number of integers. It is precisely because of Gauss's leadership that type theory became a major topic of number theory in19th century. Gauss's exposition on types and geometric expressions of types is the beginning of the so-called number geometry today. [Edit this paragraph] The role of complex numbers in number theory problems Gauss's handling of number theory problems involves complex numbers. The first is the identification of complex numbers, which is an old problem. 18 and 19 th century Many outstanding mathematicians have been asked, "What is a complex number?" I don't know. Mathematicians such as Leibniz and Euler are at a loss. Gauss unconditionally used complex numbers in proving the basic theorem of algebra. This makes the original understanding of complex numbers only from the perspective of universality of operation, and extends to the confirmation of the position of complex numbers in the proof of major algebraic problems. With his superb proof of this theorem, Gauss let the mathematical community not only look at Gauss, but also look at complex numbers. Complex number introduces number theory Gaussian. Not only that, he also introduced complex numbers into number theory and founded the theory of complex integer. In this theory, Gauss proved that complex integers have essentially the same properties as ordinary integers. Euclid proved fundamental theorem of arithmetic in ordinary integers-each integer can be uniquely decomposed into the product of prime numbers, while Gauss obtained and proved it in complex integers. As long as the four invertible elements (1, i) are not taken as different factors, this unique decomposition theorem also holds for complex numbers. Gauss also pointed out that many theorems of ordinary prime numbers, including Fermat's last theorem, may be transformed into theorems of complex numbers (extended to the field of complex numbers). [Edit this paragraph] At that time, the evaluation of arithmetic research seemed understandable to anyone who had studied ordinary algebra in middle school, but it was not at all for beginners. Few people read this book at that time. The difficulty lies not in the detailed calculation examples, but in the understanding of the theme and profound thoughts. Because this book has seven parts, people humorously call it "the book of seven seals". [Edit this paragraph] After the publication of Arithmetic Research, many young mathematicians bought this book for research. Dirichlet (1805- 1859) is one of them. Dirichlet is a famous German mathematician who has made many contributions to analysis and number theory. He takes arithmetic research as his favorite treasure, hides it in his robe where he sticks to his chest, takes it with him everywhere, and takes it out to read whenever he has time. When you sleep at night, put it under your pillow and read a few paragraphs before going to bed. Many things happen. With this perseverance, Dirichlet finally opened the "seven seals" first. Later, he introduced and explained arithmetic research in detail in popular form, which made this difficult work gradually understood and mastered by more people. [Edit this paragraph] It has been recognized by the mathematical community. There is also a touching story between arithmetic research and Dirichlet. July 1849 16 is the 50th anniversary of Dr. Gauss's degree. The University of G? ttingen held a celebration, including an ingenious program, in which they asked Gauss to light his pipe with a manuscript from Arithmetic Research. Dirichlet was standing beside Gauss, and he was completely shocked when he saw this scene. At the last minute, he desperately grabbed this page of manuscript from the teacher and treasured it. This page of manuscript was not rediscovered until after Dirichlet's death, and editors are sorting out his manuscript. After the publication of Arithmetic Research, Lagrange once pessimistically thought that "the mine source has been dug" and mathematics was on the verge of despair. When he finished reading Arithmetic Research, he saw the dawn of hope with excitement. The 68-year-old man wrote to Gauss to express his heartfelt congratulations: "Your arithmetic research immediately made you a first-rate mathematician. In my opinion, the last chapter contains the most beautiful analysis findings. In order to find this discovery, people have made a long exploration. ..... Believe me, no one cheers for your achievements more sincerely than I do. " On this work, Moritz Cantor, a famous German mathematical historian in the19th century, once said, "Gauss once said:' Mathematics is the queen of science, and number theory is the queen of mathematics.' If this is true, we can add that arithmetic research is the charter of number theory. Arithmetic Research is a masterpiece of Gauss's life. When Gauss talked about this book in his later years, he said: "Arithmetic research is the wealth of history." [Edit this paragraph] Gauss's achievements Gauss originally planned to continue writing the second volume of Arithmetic Research, but this plan failed to be realized due to the change of work and the transfer of research interest. Many of Gauss's mathematical achievements were discovered only after his death. From March 30th, 1796, Gauss made a regular polygon of 17 with a ruler, and began to write a scientific diary for a long time until July 9th, 18 14. Gauss's scientific diary was borrowed from Gauss's grandson by the Royal Society of G? ttingen in 1898 to study Gauss. Since then, the contents of this scientific diary have been spread to 43 years after Gauss's death. This diary *** 146 research results, because only for personal use, so each record is often written in a few words, very short. Some projects are so simple that even experts are confused. 1796, 10, 1 1, and Vicki Musger Gump 1799 are still a mystery. In July 1796, there was such a diary: EYPHKA! Num=△+△+△ EYPHKA means discovery in Greek. At that time, Archimedes suddenly discovered the law of buoyancy while taking a bath. He jumped out of the bathtub excitedly and ran wildly in the street, shouting "Efka!" " Gauss found the proof of a difficult theorem put forward by Fermat here: every positive integer is the sum of three triangular numbers. Once Gauss's scientific diary was published, it caused a sensation in the whole scientific community. People learned for the first time that many important achievements were actually discovered by Gauss a long time ago, but they were published very late, and some were not even published at all. It was not until the diary was published that people knew the double periodicity of elliptic function, so that this great achievement fell asleep in the diary 100 years. A diary of1March 9 1797 clearly shows that Gauss discovered this achievement; Later, there was another one, which showed that Gauss further realized the general double periodicity. This problem was independently developed by Jacobi (1804- 185 1) and Abel, and later became the core of19th century function theory. Similar examples are too numerous to mention. So many important discoveries have been buried in the diary for decades or even a century! Faced with this incredible fact, mathematicians were shocked. If these contents are published in time, it will undoubtedly bring unprecedented honor to Gauss, because any achievements in the diary were world-class at that time. If these contents are published in time, later mathematicians can be prevented from struggling in many important fields, and the history of mathematics will be greatly rewritten. Some mathematicians estimate that the development of mathematics now may be more advanced than half a century. [Edit this paragraph] Why did this happen in the social environment and Gauss's personal character at that time? This has a very important relationship with the social environment at that time and Gauss's personal character. /kloc-in the 0/8th century, there was a fierce debate in the field of mathematics. Mathematicians hold their own opinions and blame each other. Due to the lack of strict argument, various mistakes appeared in the debate. In order to prove their arguments, they often brag and satirize each other, which left a deep impression on Gauss. Although Gauss was born in poverty, like his parents, he had a strong self-esteem and was extremely cautious about scientific research, which made him not publish this diary before his death. He believes that these research results need further argumentation. His motto in scientific research is "less is better than more". Gauss's rigorous academic attitude made later scientists pay a huge price, but it also brought benefits to scientific research. Gauss's published works are still as correct and important as the first publication. His publication is code, which is better than other human codes, because it will not be found to have any problems whenever and wherever. Gauss's attitude towards learning is like a motto in King Lear. He wrote neatly under his portrait: "Nature, you are my goddess, and my service in this life must obey your law." Gauss's achievements in the field of mathematics are enormous. Later, people asked him the secret of his success, and he replied with his unique humility: "If others think about the truth of mathematics as deeply and persistently as I do, he will also find my discovery." In order to prove his conclusion, he once pointed to a topic on page 633 of Arithmetic Research and said emotionally: "People say I am a genius, don't believe it! You see, this question only takes a few lines, but it took me four years. I haven't thought about its symbol for almost a week in four years. " You can refer to this website:/resource/book/edu/kpts/joy02010/0003 _ ts086011.htm.