Pierre de Fermat (August 17, 1601 - January 12, 1665) was a 17th-century French lawyer and an amateur mathematician. It is called amateur because Pierre de Fermat had a full-time job as a lawyer. Based on the actual French pronunciation and with reference to the English pronunciation, his surname is also often translated as "Ferma". Fermat's Last Theorem is customarily called Fermat's Last Theorem in China. The original name in Western mathematics circles is "Last", which means: all other conjectures have been confirmed, and this is the last . The famous mathematics historian E.T. Bell called Pierre de Fermat "the king of amateur mathematicians" in his book written in the early 20th century. Bell was convinced that Fermat was more accomplished than most of the professional mathematicians of his generation. The 17th century was a century when outstanding mathematicians were active, and Bell believed that Fermat was the most prolific star among mathematicians in the 17th century.
Fermat never received any specialized mathematical education in his life , and mathematical research was nothing more than a hobby. However, there was no mathematician in France in the 17th century who could rival him: he was one of the inventors of analytic geometry; his contribution to the birth of calculus was second only to Isaac Newton. , Gottfried Wilhelm van Leibniz, he was also the main founder of probability theory, and the person who single-handedly supported the world of number theory in the 17th century. In addition, Fermat also made important contributions to physics. Fermat, a generation of mathematical genius, can be called the greatest French mathematician in the 17th century.
Pierre de Fermat (1601-1665)
Childhood life
On August 17, 11601, Fermat was born in Beaumont de Lomagne, near Toulouse in southern France. His father, Dominique Fermat, opened a large leather store in the local area and owned a very rich industry. His mother, Claire de Longe, was born into a family of judges. Fermat's father was respected by people because of his wealth and business management, and thus received the title of second consul of the Beaumont-Lomen region. Dominic The great wealth and Rogge's great aristocracy contributed to Fermat's extremely wealthy status. However, when he was young, Fermat did not have much sense of superiority because of his family's wealth.
Due to his wealthy family, his father specially hired two tutors for Pierre de Fermat. He did not go to school but received systematic education at home. One of them was his uncle Pierre. Fermat received a good enlightenment education , which cultivated his wide range of interests and hobbies , and also had an important impact on his character. Although Fermat was not a child prodigy as a child, he was quite smart. Fermat's father was more liberal and did not dote on his children, so Fermat studied very hard and did well in liberal arts and science. However, Pierre de Fermat's favorite subject was still mathematics. 14 When he was 16 years old, Fermat entered the École Beaumont de Lomagne. In 11617, Fermat was preparing to take the university entrance examination. His father wanted him to study law. Fermat also liked this subject, so there was not much controversy. He accepted his father's arrangement. Later he studied law at the University of Orleans and the University of Toulouse.
Official career
In France in the 117th century, the most important profession for men was to be a lawyer. Studying law became fashionable and enviable. Interestingly, France has created good conditions for those "quasi-lawyers" who have property but lack qualifications to become lawyers as soon as possible. In 1523, King Francois I organized the establishment of a special agency to sell official titles and titles, and publicly sold official positions. Once this social phenomenon of selling official positions arose, it became out of control in response to the needs of the times.
Selling official positions not only caters to the wealthy, allowing them to obtain official positions and thus improve their social status, but also improves the government's financial situation. Therefore, by the 17th century, any official position other than palace officials and military officers could be bought and sold. To this day, the positions of court clerk, notary , and conveyor have not completely shed their transactional nature. France's specialty of buying official positions has benefited many middle class people, and Fermat was no exception. Before Fermat graduated from university, he bought the positions of "lawyer" and "senator" in Beaumont de Lomagne. In 1631, after Fermat returned to his hometown after graduating, he easily became a member of the Toulouse Parliament.
Although Fermat did not lose his official position from the time he entered society until his death, and was promoted year by year, according to records, Fermat did not have any political achievements, and his ability to deal with officialdom was also very average, let alone any leadership skills. However, Fermat did not interrupt his promotion. After seven years as a member of the local parliament, Fermat was promoted to an investigative senator, an official position with the power to investigate and question the executive.
In 11642, there was an authoritative person named Briasias, who was an adviser to the Supreme Court. Briseas recommended Fermat to enter the Supreme Criminal Court and the main court of the French Grand Council, which gave Fermat better opportunities for promotion in the future. In 1646, Fermat was promoted to chief speaker of the Parliament, and later served as chairman of the Catholic League. Fermat's official career did not have any outstanding achievements worthy of praise, but Fermat never used his power to extort people, never took bribes, was honest, open and honest, and won people's trust and praise.
Family life
Pierre de Fermat became a petition committee member in the same place when he was 30 years old. In the same year, he married Louise Long and had three sons and two daughters, one of whom was Clement Shan. Clement Samuel Fermat (Clement Samuel Fermat) (Little Fermat) became Pierre de Fermat's main assistant in scientific research, and after Fermat's death, he compiled and published Pierre de Fermat work results. In fact, this publication is the source of what is now known as Fermat's Last Theorem.
Fermat discovered the basic principles of analytic geometry independently of René Descartes
Before 11629, Fermat began to rewrite the lost "" of the third century BC ancient Greek geometer Apollonius Plane trajectory book. He used algebraic methods to supplement some lost proofs of Apollonius on trajectories, summarized and organized ancient Greek geometry, especially Apollonius' theory of conics, and made general research on curves. In 1630, he wrote an eight-page paper "Introduction to Plane and Three-dimensional Trajectories" in Latin.
Fermat began to correspond with Marin Mersenne and Roberval, the great mathematicians at that time, in 1636. Fermat talked a little about his mathematical work. However, "Introduction to Plane and Solid Trajectories" was published 14 years after Fermat's death. Therefore, before 1679, few people knew about Fermat's work, but now it seems that Fermat's work is groundbreaking. of.
"Introduction to Plane and Three-dimensional Trajectories" explains Fermat's discovery. He pointed out: "An equation determined by two unknown quantities corresponds to a trajectory and can draw a straight line or curve." Fermat's discovery was seven years earlier than René Descartes's discovery of the basic principles of analytic geometry. Year. In the book, Fermat also discussed the equations of general straight lines and circles, as well as hyperbola, ellipse, and parabola.
Descartes looked for its equation from a trajectory, while Fermat studied the trajectory starting from the equation. These are two opposite aspects of the basic principles of analytic geometry.
In a letter in 1643, Fermat also talked about his ideas in analytic geometry. He talked about cylindrical , elliptical paraboloid, double-leaf hyperboloid and ellipsoid , and pointed out that an equation containing three unknown quantities represents a curved surface, and did further research on this.
Contribution to Probability Theory
As early as the ancient Greek period, contingency and inevitability and their relationship have aroused the interest and debate of many philosophers, but the mathematical description and treatment of them was only after the 15th century. In the early 16th century, mathematicians such as Cardano appeared in Italy to study the game opportunities in dice and explore the division of gambling money at the points of the game. In the 17th century, French Pascal and Fermat studied the Italian work "Abstract" by Pacioli and established communication links, thereby establishing the basis of probability.
The discussion in the letters between Fermat and Pascal in 11654 can be regarded as the beginning of probability theory.
In 11656, his communication with Christian Huygens, the official founder of probability theory, made Huygens increase his interest in probability.
Fermat and Blaise Pascal established the basic principles of probability theory - the concept of mathematical expectation - in their correspondence and writings. It starts with a mathematical problem: how to determine the division of stakes in an interrupted game between players of assumed equal skill, given the scores of the two players at the time of the interruption and their scores at the time of the interruption. The number of points required to win the game. Fermat discussed this: a situation where player A needs 4 points to win and player B needs 3 points to win. This is Fermat's solution to this special situation. Because apparently up to four times can decide the outcome.
The concept of general probability space is a thorough axiomaticization of people's intuitive ideas about concepts. From a purely mathematical point of view, finite probability spaces seem mundane. But once random variables and mathematical expectations are introduced, they become a magical world. This is Fermat's contribution.
's contribution to calculus The tangent problem of the
curve and the maximum and minimum value problems of the function are one of the origins of calculus. This work is relatively old, dating back to ancient Greece. Archimedes used the exhaustive method to find the area of any figure enclosed by a curve. Because the exhaustion method was cumbersome and cumbersome, it was gradually forgotten and was not taken seriously until the 16th century. Since Johannes Kepler encountered the problem of how to determine the area and arc length of an ellipse when he was exploring the laws of planetary motion, the concepts of infinity and infinitesimal were introduced and replaced the cumbersome exhaustion method. Although this method is not perfect, it has opened up a very broad thinking space for mathematicians from Cavalieri to Fermat.
Fermat established the methods of finding tangents, finding maximum and minimum values, and definite integrals, and made a significant contribution to calculus.
Contribution to Number Theory
In the early 117th century, ancient Greek mathematicians from the third century AD were circulated in Europe. The book Arithmetic written by Diophantus. In 1621, Fermat bought this book in Paris, and he used his spare time to conduct in-depth research on the indefinite equations in the book. Fermat limited the study of indefinite equations to the range of integers, thus beginning the branch of mathematics known as number theory.
Fermat's achievements in the field of number theory are huge, the main ones are:
Fermat's Last Theorem
In 11637, when Fermat was reading the Latin translation of Diophantus's "Arithmetic", he once wrote the 8th proposition in Volume 11. Next to it he wrote: " divides a cubic number into the sum of two cubic numbers, or a fourth power into the sum of two fourth powers, or generally divides a power higher than the second into two powers of the same power. sum, this is impossible. I'm sure I've discovered a wonderful proof of this, but the space here is too small to write . ”
Fermat’s last theorem is expressed as : n2 is an integer, then the equation x^n+y^n=z^n does not have an integer solution that satisfies xyz≠0. This is an indefinite equation, which has been solved by the British mathematician Wile Si proved (in 1995), the proof process is quite difficult!
Fermat's little theorem
a^p-a≡0 (mod p), where p is a prime number and a is a positive integer. In fact, its proof is relatively simple. It is a special case of Euler's theorem. Euler's theorem is: a^φ(n)-1≡0(mod n), a and n are both positive integers, and φ(n) is the Euler function, which means that if it is relatively prime with n, it is less than n. The number of positive integers.
In addition:
(1) All prime numbers greater than 2 can be divided into two forms: 4n+1 and 4n+3.
(2) Prime numbers of the form 4n+1 can be, and only It can be expressed as the sum of two square numbers .
(3) There is no prime number of the form 4n+3, which can be expressed as the sum of two square numbers
(4) of the form 4n+1. A prime number can and can only be the hypotenuse of a right triangle with an integer side; the square of 4n+1 is and can only be the hypotenuse of two such right triangles; similarly, the mth power of 4n+1 is and It can only be the hypotenuse of m such right triangles.
(5) The area of a right triangle with a rational number cannot be a square number.
(6) The prime number of the 4n+1 shape can only be its square. It can be expressed in one way as the sum of two square numbers; its third and fourth powers can only be expressed in two ways as the sum of two square numbers; its fifth and sixth powers can only be expressed in three ways. is the sum of two square numbers, and so on, until infinity.
- discovered the second pair of affinity numbers: 17296 and 18416. In the sixteenth century, some people believed that there was only one pair of affinity numbers in natural numbers: 220 and 284. . Some boring people even add superstition or mystery to the affinity numbers, and make up many myths and stories to promote the pairing of affinity numbers in magic, magic, astrology, and divination. It plays an important role, etc.
More than 2,500 years after the birth of the first pair of affinity numbers, in 1636, Fermat found the second pair of affinity numbers, 17296 and 18416, rekindling the search for affinity numbers and finding light in the darkness. Two years later, the French mathematician René Descartes, the "father of analytic geometry", also announced on March 31, 1638 that he had found the third pair of affinity numbers, 9437056 and 9363584. In two years, Fermat and Descartes broke the silence of more than two thousand years and sparked a wave of renewed search for affinity numbers in the mathematical community.
Fermat's works
Contribution to optics
Fermat's outstanding contribution in optics is to propose the principle of minimum action , also called the principle of shortest time action . This principle has a long history. As early as the ancient Greek period, Euclid proposed the law of linear propagation of light and the law of reflection. Later, Helen revealed the theoretical essence of these two laws - light takes the shortest path. After several years, this law was gradually expanded into a natural law, and then became a philosophical concept. A more general conclusion that " nature acts in the shortest possible way " was eventually drawn and influenced Fermat. Fermat's brilliance lies in turning this philosophical concept into a scientific theory.
Fermat also discussed the situation when light travels in a medium that changes point by point, and its path takes a very small curve. And explained some problems using the principle of least action. This gives great encouragement to many mathematicians. In particular, Leonhard Euler actually used variation method technique to apply this principle to find the extreme value of a function.This directly led to the achievement of Lagrangian , which gave the specific form of the principle of minimum action: for a particle , the integral of the product of its mass, speed and the distance between two fixed points is Maximum and minimum values; that is, for the actual path taken by the particle, it must be maximum or minimum.
Fermat's Principle
Passed away
Fermat was in good health throughout his life, but nearly died in the plague of 1652. After New Year's Day in 1665, Fermat began to feel physical changes, so he suspended his duties on January 10. On the third day, Fermat died. Fermat was buried in Castres Cemetery and later in the family cemetery in Toulouse.
Fermat’s Monument
