Pierre de Fermat ( fr. Pierre de Fermat , August 17, 1601 - January 12, 1665 ) is a French self-taught mathematician , one of the creators of analytical geometry , mathematical analysis , probability theory and number theory . A lawyer by profession, since 1631 - Advisor to the Parliament in Toulouse . Brilliant polyglot . Most famous for the formulation of Fermat’s Great Theorem , “the most famous mathematical riddle of all time” [3] .
| Pierre de Fermat | |
|---|---|
| Pierre de fermat | |
 | |
| Date of Birth | August 17, 1601 |
| Place of Birth | Beaumont de Loman |
| Date of death | January 12, 1665 (63 years old) |
| Place of death | Castres |
| A country | |
| Scientific field | maths |
| Place of work | |
| Alma mater | University of Toulouse |
| Academic degree | ( 1626 ) |
| Known as | author of Fermat's Great Theorem |
Biography
Pierre Fermat was born on August 17, 1601 in the Gascon town of Beaumont de Lomagne ( Beaumont-de-Lomagne , France ). His father, Dominic Fermat, was a prosperous tanner, a second city consul. In the family, besides Pierre, there was another son and two daughters. The farm received a law degree - first in Toulouse (1620-1625), and then in Bordeaux and Orleans (1625-1631).
In 1631, having successfully completed his studies, Fermat redeemed the position of royal adviser to the parliament (in other words, a member of the high court) in Toulouse. In the same year, he married a distant relative of his mother, Louise de Long. They had five children [4] .
Fast career growth allowed Fermat to become a member of the Chamber of Edicts in the city of Castres (1648). It is to this position that he owes the addition to his name of a sign of nobility - particles de ; from this time he becomes Pierre de Fermat .
The calm, measured life of a provincial lawyer left Fermat time for self-education and mathematical research. In 1636, he wrote a treatise, Introduction to the Theory of Plane and Spatial Places, where, independently of Descartes’s “ Geometry ” (published a year later), he presented analytic geometry . In 1637 he formulated his “ Great Theorem ”. In 1640, he announced the less famous, but much more fundamental Small Fermat's theorem . He conducted active correspondence (through Maren Mersenne ) with prominent mathematicians of that period. From his correspondence with Pascal begins the formation of ideas of probability theory .
In 1637, the conflict between Fermat and Descartes began. Fermat spoke destructively of the Cartesian Diopter, Descartes did not remain in debt, gave a crushing review of Fermat's work on analysis, and hinted that some of Fermat's results were plagiarized from the Cartesian Geometry . Fermat’s method for conducting tangents Descartes did not understand (the presentation in Fermat’s article was really short and careless) and, as a call, suggested that the author find the tangent to the curve, later called the “ Cartesian leaf ”. Fermat was not slow to give two correct solutions - one according to Fermat's article, the other based on the ideas of Descartes' Geometry ", and it became clear that the Fermat method is simpler and more convenient. Gerard Desargues acted as a mediator in the dispute - he acknowledged that the Fermat method is universal and correct in essence, but presented unclear and incomplete. Descartes apologized to the opponent, but until the end of his life he treated Fermat unfriendly [5] .
Around 1652, Fermat had to refute the report of his demise during the plague epidemic; he really got infected, but survived, and the death of many colleagues promoted Fermat to the post of supreme parliamentary judge. In 1654, Fermat made the only long-distance journey in his life in Europe. In 1660, his meeting with Pascal was planned, but due to the poor health of both scientists, the meeting did not take place [4] .
Pierre de Fermat died on January 12, 1665 in the city of Castres , during a visiting court session. Initially, he was buried there, in Castres, but later (1675) the ashes were transferred to the family tomb of Fermat in the Toulouse church of the Augustinians. During the French Revolution, the remains of Fermat were lost.
The oldest son of the scientist, Clement-Samuel (also a lover of mathematics), published in 1670 a posthumous collection of his father's works (several hundred letters and notes), from which the scientific community learned about the remarkable discoveries of Pierre Fermat. Additionally, he published "Comments on Diophantus", made by his father in the margins of the translation of Diophantus' book; from this moment the fame of Fermat's Great Theorem begins [6] .
Contemporaries characterize Fermat as an honest, accurate, balanced and affable person, brilliantly erudite both in mathematics and in the humanities, an expert on many ancient and living languages in which he wrote good poems [7] .
Scientific activity
Fermat’s discoveries came to us through a collection of his extensive correspondence (mainly through Mersenne ), published posthumously by the son of a scientist. Fermat gained fame as one of the first mathematicians in France, although he did not write books (there were no scientific journals yet), limiting himself to letters to colleagues. Among his correspondents were Rene Descartes , Blaise Pascal , Gerard Desargues , Gilles Roberval , John Wallis and others. Fermat's only work, published in print during his lifetime, was A Treatise on Straightening (1660), which was published as an appendix to the work of his fellow countryman and friend Antoine de Lalouver and (at Fermat's request) without indicating the name of the author.
Unlike Descartes and Newton , Fermat was a pure mathematician - the first great mathematician in new Europe. Regardless of Descartes, he created analytic geometry . Newton used to be able to use differential methods to conduct tangents , find highs and calculate areas. True, Fermat, unlike Newton, did not bring these methods into the system, but Newton later admitted that it was Fermat's work that prompted him to create an analysis [8] .
The main merit of Pierre Fermat is the creation of number theory .
Number Theory
Mathematicians of Ancient Greece since the time of Pythagoras have collected and proved various assertions related to natural numbers (for example, methods for constructing all Pythagorean triples , a method for constructing perfect numbers , etc.). Diophantus of Alexandria (3rd century A.D.) in his "Arithmetic" considered numerous problems of solving in rational numbers algebraic equations with several unknowns (now called Diophantine equations that need to be solved in integers). This book (not fully) became known in Europe in the 16th century , and in 1621 it was published in France and became Fermat's handbook.
The farm was constantly interested in arithmetic problems, exchanged complex problems with contemporaries. For example, in his letter, called the “Second Call to Mathematicians” (February 1657), he proposed finding a general rule for solving the Pell equation in integers. In the letter, he suggested finding solutions at a = 149, 109, 433. A complete solution to the Fermat problem was found only in 1759 by Euler .
Fermat began with problems about magic squares and cubes, but gradually switched to the laws of natural numbers - arithmetic theorems. There is no doubt the influence of Diophantus on Fermat, and it is symbolic that he writes down his amazing discoveries in the fields of "Arithmetic."
Fermat discovered that if a is not divisible by a prime p , then the number always divisible by p (see Fermat’s Little Theorem ). Euler later gave a proof and generalization of this important result: see Euler's Theorem .
Finding that the number prime for k ≤ 4, Fermat decided that these numbers are prime for all k, but Euler subsequently showed that for k = 5 there is a divisor 641. It is still unknown whether the Fermat primes are finite or infinite.
Euler proved (1749) another Fermat's conjecture (Fermat himself rarely gave evidence of his statements): primes of the form 4 k +1 are represented as the sum of squares (5 = 4 + 1; 13 = 9 + 4), and in a unique way, and for numbers containing, in their factorization, primes of the form 4 k +3 to an odd degree, such a representation is impossible. For Euler, this proof was worth 7 years of labor; Fermat himself proved this theorem indirectly, invented by him inductive "method of infinite descent . " This method was published only in 1879; however, Euler reconstructed the essence of the method from several remarks in Fermat's letters and repeatedly applied it successfully. Later, an improved version of the method was used by Poincare and Andre Weil .
Fermat developed a method for systematically finding all divisors of a number, formulated a theorem on the possibility of representing an arbitrary number with a sum of not more than four squares ( Lagrange's theorem on the sum of four squares ). His most famous statement is Fermat 's Great Theorem (see below).
Ferm's figures were of great interest. In 1637, he formulated the so-called “golden theorem” [9] :
- Every natural number is either triangular or the sum of two or three triangular numbers.
- Every natural number is either a square or the sum of two, three or four square numbers ( Lagrange's theorem on the sum of four squares ).
- Every natural number is either pentagonal , or the sum of two to five pentagonal numbers.
- Etc.
Many outstanding mathematicians were involved in this theorem; Cauchy was able to give a complete proof in 1813 [10] .
Many witty methods used by Fermat remained unknown. Once Mersenne asked Fermat to find out if the multi-digit number 100 895 598 169 is prime. The farm was not slow to report that he did not explain how he found these dividers.
Fermat's arithmetic discoveries were ahead of time and were forgotten for 70 years until Euler became interested in them, who published a systematic number theory. One reason for this is that the interests of most mathematicians have switched to mathematical analysis ; it was probably due to the fact that Fermat used obsolete and cumbersome mathematical symbolism of Viet instead of much more convenient notation of Descartes [11] .
Mathematical Analysis and Geometry
Fermat almost by modern rules found tangents to algebraic curves . It was these works that prompted Newton to create an analysis [8] . In textbooks on mathematical analysis, one can find Fermat's important lemma , or a necessary sign of an extremum : at the points of an extremum, the derivative of a function is equal to zero.
Fermat formulated the general law of differentiation of fractional degrees. He gave a general way for drawing tangents to an arbitrary algebraic curve . In A Treatise on Quadratures (1658), Fermat showed how to find the area under hyperbolas of various degrees by extending the formula for integrating the degree even in cases of fractional and negative exponents. In A Treatise on Straightening, Fermat described a general way to solve the most difficult problem of finding the length of an arbitrary (algebraic) curve.
Along with Descartes , Fermat is considered the founder of analytic geometry . In the introduction to the theory of planar and spatial places, which became known in 1636, he was the first to classify curves according to the order of their equations, to establish that a first-order equation defines a straight line and a second-order equation a conic section . Developing these ideas, Fermat went further than Descartes and tried to apply analytic geometry to space, but did not significantly advance in this topic.
Other Achievements
Regardless of Pascal, Fermat developed the foundations of probability theory . It is from the correspondence of Fermat and Pascal ( 1654 ), in which they, in particular, came to the concept of mathematical expectation and the theorems of addition and multiplication of probabilities, this remarkable science counts its history. The results of Fermat and Pascal were presented in Huygens 's book On Calculations in a Gambling (1657), the first guide to probability theory.
Fermat’s name bears the basic variational principle of geometric optics , by virtue of which light in an inhomogeneous medium chooses the path that takes the least time (however, Fermat believed that the speed of light is infinite, and formulated the principle more foggy). With this thesis begins the history of the main law of physics - the principle of least action .
Fermat transferred to the three-dimensional case (internal tangency of spheres) the Vieta algorithm for the Apollonius problem of tangency of circles [12] .
The Great Farm Theorem
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Fermat is widely known for the so-called great (or last) Fermat's theorem . The theorem was formulated by him in 1637 , on the sidelines of Diophantus's book “Arithmetic,” with a note that the witty proof he found of this theorem was too long to be given in the margins.
Most likely, his proof was not correct, since later he published the proof only for the case . The proof, developed in 1994 by Andrew Wiles , contains 129 pages and was published in the journal Annals of Mathematics in 1995 .
The simplicity of the formulation of this theorem attracted many amateur mathematicians, the so-called fermatists . Even after Wiles’s decision, letters are sent to all academies of sciences with “proofs” of Fermat’s great theorem.
Perpetuation of memory
- In 1935, the International Astronomical Union named Pierre Fermat a crater on the visible side of the moon .
- The Fermat Mathematical Prize has been awarded since 1989.
- In Toulouse, the name Fermat was given to the street, as well as the oldest and most prestigious lyceum of Toulouse ( Lycée Pierre de Fermat ).
- In Beaumont de Lomani, where Fermat was born, his museum is open and a monument to the scientist is erected. The street and the hotel located on this street are named in his honor.
- In honor of Fermat, various mathematical theorems and concepts are named, including:
- Fermat's Great Theorem
- Fermat's Little Theorem
- Fermat factorization method
- Steiner problem
- Spiral Farm
- Farm Point
- Farm Numbers
- Fermat's Little Theorem
- Fermat's Great Theorem
Farm in Fiction and Stamps
Alexander Kazantsev wrote the sci-fi novel hypothesis “Bubbling Emptiness”. The first book of this novel, “Sharper the Swords,” is dedicated to describing the life and achievements of Pierre Fermat.
In the year of the scientist’s 400th anniversary (2001), the French Post issued a postage stamp (0.69 euros) with his portrait and the formulation of the Great Theorem.
Notes
- ↑ http://www.nytimes.com/1983/07/19/science/german-is-hailed-in-math-advance.html
- ↑ Archive for the history of mathematics MacTyutor
- ↑ Alvarez, 2015 , p. 15.
- ↑ 1 2 Stillwell D. Mathematics and its history. - Moscow-Izhevsk: Institute for Computer Research, 2004, pp. 211-212.
- ↑ Alvarez, 2015 , p. 124-128.
- ↑ Alvarez, 2015 , p. 40.
- ↑ Bell E. T. Creators of Mathematics, 1979 , p. 58.
- ↑ 1 2 Vavilov S.I. Isaac Newton. 2nd supplemented edition. M.-L .: Ed. USSR Academy of Sciences, 1945, chapter 13.
- ↑ Matvievskaya G.P. The doctrine of number in the medieval Near and Middle East. - Tashkent: FAN, 1967. - S. 22-23. - 344 p. .
- ↑ Vilenkin N. Ya. Popular combinatorics. - M .: Nauka, 1975 .-- S. 10-11. - 208 p.
- ↑ Alvarez, 2015 , p. 91.
- ↑ Barabanov O. O., Barabanova L. P. Algorithms for solving the navigation difference-range-finding problem - from Apollonius to Cauchy // History of Science and Technology, 2008, No. 11, S.2-21.
Proceedings in the Russian translation
- Fermat P. Studies in number theory and Diophantine analysis. M .: Nauka, 1992. ISBN 5-02-014121-6 . Reprints: 2007, 2015.
Literature
- Alvarez L.F. A. The most difficult task in the world. Farm. The Great Fermat Theorem // Science. The greatest theories. - M .: De Agostini, 2015 .-- Issue. 18 . - ISSN 2409-0069 .
- Bashmakova I.G. Diophantus and Fermat (on the history of the method of tangents and extrema). Historical and mathematical research , 17, 1966, S. 185-207.
- Bashmakova I. G., Slavutin E. I. The history of the Diophantine analysis from Diophantus to Fermat. M .: Nauka, 1984.
- Bell E. T. Creators of Mathematics. - M .: Education, 1979. - 256 p.
- Van der Waerden B. L. Correspondence between Pascal and Fermat on probability theory. IMI, 21, 1976, S. 228-232.
- Mathematics of the XVII century // History of mathematics / Edited by A.P. Yushkevich , in three volumes. - M .: Nauka, 1970 .-- T. II.
- Farm // Brockhaus and Efron Encyclopedic Dictionary : in 86 volumes (82 volumes and 4 additional). - SPb. , 1890-1907.
- Freiman L. S. Fermat, Torricelli, Roberval. In the book: At the origins of classical science. M .: Nauka, 1968, S. 173-254.
- Hramov Yu. A. Fermat Pierre (Pierre de Fermat) // Physicists: Biographical Reference / Ed. A.I. Akhiezer . - Ed. 2nd, rev. and add. - M .: Nauka , 1983 .-- S. 275 .-- 400 p. - 200,000 copies. (in per.)
- Shawl . A historical review of the origin and development of geometric methods . Ch. 2, § 10-14. M., 1883.
Links
- John J. O'Connor and Edmund F. Robertson . Fermat, Pierre (English) - biography in the MacTutor archive.
- Fermat's achievements
- The Life and times of Pierre de Fermat (1601-1665) from WW Rouse Ball's History of Mathematics
- The Mathematics of Fermat's Last Theorem