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149-419: Marie-Sophie Germain ( French: [maʁi sɔfi ʒɛʁmɛ̃] ; 1 April 1776 – 27 June 1831) was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's library, including ones by Euler , and from correspondence with famous mathematicians such as Lagrange , Legendre , and Gauss (under

298-435: A {\displaystyle c/a} , presumably for actual use as a "table", for example, with a view to applications. It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy , for example, truly came into its own only later. It has been suggested instead that the table was a source of numerical examples for school problems. While evidence of Babylonian number theory

447-652: A 1 mod m 1 {\displaystyle n\equiv a_{1}{\bmod {m}}_{1}} , n ≡ a 2 mod m 2 {\displaystyle n\equiv a_{2}{\bmod {m}}_{2}} could be solved by a method he called kuṭṭaka , or pulveriser ; this is a procedure close to (a generalisation of) the Euclidean algorithm, which was probably discovered independently in India. Āryabhaṭa seems to have had in mind applications to astronomical calculations. Brahmagupta (628 AD) started

596-491: A Diophantine equations a polynomial equations to which rational or integer solutions are sought. While Greek astronomy probably influenced Indian learning, to the point of introducing trigonometry , it seems to be the case that Indian mathematics is otherwise an indigenous tradition; in particular, there is no evidence that Euclid's Elements reached India before the 18th century. Āryabhaṭa (476–550 AD) showed that pairs of simultaneous congruences n ≡

745-574: A brain hemorrhage . Jacob von Staehlin  [ de ] wrote a short obituary for the Russian Academy of Sciences and Russian mathematician Nicolas Fuss , one of Euler's disciples, wrote a more detailed eulogy, which he delivered at a memorial meeting. In his eulogy for the French Academy , French mathematician and philosopher Marquis de Condorcet , wrote: il cessa de calculer et de vivre — ... he ceased to calculate and to live. Euler

894-518: A convex polyhedron , and hence of a planar graph . The constant in this formula is now known as the Euler characteristic for the graph (or other mathematical object), and is related to the genus of the object. The study and generalization of this formula, specifically by Cauchy and L'Huilier , is at the origin of topology . Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of

1043-557: A bust commissioned by the Paris City Council. In January 2020, Satellogic , a high-resolution Earth observation imaging and analytics company, launched a ÑuSat type micro-satellite named in honor of Sophie Germain. E. Dubouis defined a sophien of a prime n to be a prime θ where θ  =  kn  + 1 , for such n that yield θ such that x  =  y  + 1 (mod θ ) has no solutions when x and y are prime to n . A Sophie Germain prime

1192-416: A candidate for its presidency by Jean le Rond d'Alembert , Frederick II named himself as its president. The Prussian king had a large circle of intellectuals in his court, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs. He was, in many ways,

1341-498: A disciple of Theodorus's; he worked on distinguishing different kinds of incommensurables , and was thus arguably a pioneer in the study of number systems . (Book X of Euclid's Elements is described by Pappus as being largely based on Theaetetus's work.) Euclid devoted part of his Elements to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic to it (Books VII to IX of Euclid's Elements ). In particular, he gave an algorithm for computing

1490-562: A dissertation that compared the philosophies of René Descartes and Isaac Newton . Afterwards, he enrolled in the theological faculty of the University of Basel. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono with which he unsuccessfully attempted to obtain a position at the University of Basel. In 1727, he entered the Paris Academy prize competition (offered annually and later biennially by

1639-502: A family friend, requesting that he ensure Gauss's safety. General Pernety sent the chief of a battalion to meet with Gauss personally to see that he was safe. As it turned out, Gauss was fine, but he was confused by the mention of Sophie's name. Three months after the incident, Germain disclosed her true identity to Gauss. He replied: How can I describe my astonishment and admiration on seeing my esteemed correspondent M. Le Blanc metamorphosed into this celebrated person ... when

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1788-527: A female scientist". When Lagrange saw the intelligence of M. Le Blanc, he requested a meeting, and thus Sophie was forced to disclose her true identity. Fortunately, Lagrange did not mind that Germain was a woman, and he became her mentor. Germain first became interested in number theory in 1798 when Adrien-Marie Legendre published Essai sur la théorie des nombres . After studying the work, she opened correspondence with him on number theory, and later, elasticity . Legendre included some of Germain's work in

1937-455: A first foray towards both Évariste Galois 's work and algebraic number theory . Starting early in the nineteenth century, the following developments gradually took place: Algebraic number theory may be said to start with the study of reciprocity and cyclotomy , but truly came into its own with the development of abstract algebra and early ideal theory and valuation theory; see below. A conventional starting point for analytic number theory

2086-771: A house in Charlottenburg , in which he lived with his family and widowed mother. Euler became the tutor for Friederike Charlotte of Brandenburg-Schwedt , the Princess of Anhalt-Dessau and Frederick's niece. He wrote over 200 letters to her in the early 1760s, which were later compiled into a volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess . This work contained Euler's exposition on various subjects pertaining to physics and mathematics and offered valuable insights into Euler's personality and religious beliefs. It

2235-529: A keen interest in mathematics, and distinguished clearly between arithmetic and calculation. (By arithmetic he meant, in part, theorising on number, rather than what arithmetic or number theory have come to mean.) It is through one of Plato's dialogues—namely, Theaetetus —that it is known that Theodorus had proven that 3 , 5 , … , 17 {\displaystyle {\sqrt {3}},{\sqrt {5}},\dots ,{\sqrt {17}}} are irrational. Theaetetus was, like Plato,

2384-642: A medic in the Russian Navy . The academy at Saint Petersburg, established by Peter the Great , was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy's benefactress, Catherine I , who had continued the progressive policies of her late husband, died before Euler's arrival to Saint Petersburg. The Russian conservative nobility then gained power upon

2533-407: A naïve sense, analogous to primes among the integers.) The initial impetus for the development of ideal numbers (by Kummer ) seems to have come from the study of higher reciprocity laws, that is, generalisations of quadratic reciprocity . Number fields are often studied as extensions of smaller number fields: a field L is said to be an extension of a field K if L contains K . (For example,

2682-469: A new field of study, analytic number theory . In breaking ground for this new field, Euler created the theory of hypergeometric series , q-series , hyperbolic trigonometric functions , and the analytic theory of continued fractions . For example, he proved the infinitude of primes using the divergence of the harmonic series , and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to

2831-473: A non-elementary one. Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually broad within mathematics. Analytic number theory may be defined Some subjects generally considered to be part of analytic number theory, for example, sieve theory , are better covered by

2980-442: A professional colleague, as they would any man, by simply rejecting the work". So Augustin-Louis Cauchy , who had been appointed to review her work, recommended her to publish it, and she followed his advice. One further work of Germain's on elasticity was published posthumously in 1831, her " Mémoire sur la courbure des surfaces ". She used the mean curvature in her research (see Honors in number theory ). Germain's best work

3129-404: A quarter of the combined output in mathematics, physics, mechanics, astronomy, and navigation in the 18th century. Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function and was the first to write f ( x ) to denote the function f applied to the argument x . He also introduced

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3278-596: A representative of the bourgeoisie to the États-Généraux , which he saw change into the National Assembly . It is therefore assumed that Sophie witnessed many discussions between her father and his friends on politics and philosophy. Gray proposes that after his political career, Ambroise-François became the director of a bank; in any case, the family remained well-off enough to support Germain throughout her adult life. Marie-Sophie had one younger sister, Angélique-Ambroise, and one older sister, Marie-Madeline. Her mother

3427-462: A result otherwise known as the Euclid–Euler theorem . Euler also conjectured the law of quadratic reciprocity . The concept is regarded as a fundamental theorem within number theory, and his ideas paved the way for the work of Carl Friedrich Gauss , particularly Disquisitiones Arithmeticae . By 1772 Euler had proved that 2  − 1 = 2,147,483,647 is a Mersenne prime. It may have remained

3576-460: A solution to several unsolved problems in number theory and analysis, including the famous Basel problem . Euler has also been credited for discovering that the sum of the numbers of vertices and faces minus the number of edges of a polyhedron equals 2, a number now commonly known as the Euler characteristic . In the field of physics, Euler reformulated Newton 's laws of physics into new laws in his two-volume work Mechanica to better explain

3725-626: A strong connection to the academy in St. Petersburg and also published 109 papers in Russia. He also assisted students from the St. Petersburg academy and at times accommodated Russian students in his house in Berlin. In 1760, with the Seven Years' War raging, Euler's farm in Charlottenburg was sacked by advancing Russian troops. Upon learning of this event, General Ivan Petrovich Saltykov paid compensation for

3874-476: A system of psychology and sociology, which were then just coming into existence. Her philosophy was highly praised by Auguste Comte . Two of her philosophical works, Pensées diverses and Considérations générales sur l'état des sciences et des lettres, aux différentes époques de leur culture , were published, both posthumously. This was due in part to the efforts of Lherbette, her nephew, who collected her philosophical writings and published them. Pensées

4023-627: A treatise on squares in arithmetic progression by Fibonacci —who traveled and studied in north Africa and Constantinople—no number theory to speak of was done in western Europe during the Middle Ages. Matters started to change in Europe in the late Renaissance , thanks to a renewed study of the works of Greek antiquity. A catalyst was the textual emendation and translation into Latin of Diophantus' Arithmetica . Pierre de Fermat (1607–1665) never published his writings; in particular, his work on number theory

4172-501: A way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis . He invented the calculus of variations and formulated the Euler–Lagrange equation for reducing optimization problems in this area to the solution of differential equations . Euler pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced

4321-402: A woman in mathematics (see Correspondence with Gauss ). The modern view generally acknowledges that although Germain had great talent as a mathematician, her haphazard education had left her without the strong base she needed to truly excel. As explained by Gray, "Germain's work in elasticity suffered generally from an absence of rigor, which might be attributed to her lack of formal training in

4470-528: A woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarising herself with [number theory's] knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the noblest courage, extraordinary talent, and superior genius. Gauss's letters to Olbers show that his praise for Germain was sincere. In the same 1807 letter, Germain claimed that if x n + y n {\displaystyle x^{n}+y^{n}}

4619-515: A younger brother, Johann Heinrich. Soon after the birth of Leonhard, the Euler family moved from Basel to the town of Riehen , Switzerland, where his father became pastor in the local church and Leonhard spent most of his childhood. From a young age, Euler received schooling in mathematics from his father, who had taken courses from Jacob Bernoulli some years earlier at the University of Basel . Around

Sophie Germain - Misplaced Pages Continue

4768-408: Is Dirichlet's theorem on arithmetic progressions (1837), whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable. The first use of analytic ideas in number theory actually goes back to Euler (1730s), who used formal power series and non-rigorous (or implicit) limiting arguments. The use of complex analysis in number theory comes later:

4917-508: Is a constant. After winning the academy contest, she was still not able to attend its sessions because of the academy's tradition of excluding women other than the wives of members. Seven years later this situation was transformed, when she made friends with Joseph Fourier , a secretary of the academy, who obtained tickets to the sessions for her. Germain published her prize-winning essay at her own expense in 1821, mostly because she wanted to present her work in opposition to that of Poisson. In

5066-485: Is a fixed rational number whose square root is not rational.) For that matter, the 11th-century chakravala method amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such. The grounds of the subject were set in the late nineteenth century, when ideal numbers , the theory of ideals and valuation theory were introduced; these are three complementary ways of dealing with

5215-471: Is a fragment of a table: the broken clay tablet Plimpton 322 ( Larsa, Mesopotamia , ca. 1800 BC) contains a list of " Pythagorean triples ", that is, integers ( a , b , c ) {\displaystyle (a,b,c)} such that a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} . The triples are too many and too large to have been obtained by brute force . The heading over

5364-839: Is a history of science and mathematics with Germain's commentary. In Considérations , the work admired by Comte, Germain argues that there are no differences between the sciences and the humanities . In 1829 Germain learned that she had breast cancer. Despite the pain, she continued to work. In 1831 Crelle's Journal published her paper on the curvature of elastic surfaces and "a note about finding y and z in 4 ( x p − 1 ) x − 1 = y 2 ± p z 2 {\displaystyle {\tfrac {4(x^{p}-1)}{x-1}}=y^{2}\pm pz^{2}} ". Mary Gray records: "She also published in Annales de chimie et de physique an examination of principles which led to

5513-426: Is a prime p such that 2 p  + 1 is also prime. The Germain curvature (also called mean curvature ) is ( k 1 + k 2 ) / 2 {\displaystyle (k_{1}+k_{2})/2} , where k 1 and k 2 are the maximum and minimum values of the normal curvature. Sophie Germain's identity states that for any { x , y } , Vesna Petrovich found that

5662-405: Is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes. Euler's name is associated with a large number of topics . Euler's work averages 800 pages a year from 1725 to 1783. He also wrote over 4500 letters and hundreds of manuscripts. It has been estimated that Leonhard Euler was the author of

5811-441: Is an algebraic number. Fields of algebraic numbers are also called algebraic number fields , or shortly number fields . Algebraic number theory studies algebraic number fields. Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study. It could be argued that the simplest kind of number fields (viz., quadratic fields) were already studied by Gauss, as

5960-433: Is contained almost entirely in letters to mathematicians and in private marginal notes. In his notes and letters, he scarcely wrote any proofs—he had no models in the area. Over his lifetime, Fermat made the following contributions to the field: The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur Goldbach , pointed him towards some of Fermat's work on

6109-464: Is irrational is credited to the early Pythagoreans (pre- Theodorus ). By revealing (in modern terms) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; its proof or its divulgation are sometimes credited to Hippasus , who was expelled or split from the Pythagorean sect. This forced a distinction between numbers (integers and

Sophie Germain - Misplaced Pages Continue

6258-635: Is known only because of the footnote in Legendre's treatise on number theory, where he used it to prove Fermat's Last Theorem for p  = 5 (see Correspondence with Legendre ). Germain also proved or nearly proved several results that were attributed to Lagrange or were rediscovered years later. Del Centina states that "after almost two hundred years her ideas were still central", but ultimately her method did not work. In addition to mathematics, Germain studied philosophy and psychology . She wanted to classify facts and generalize them into laws that could form

6407-850: Is little evidence that she really understood the Disquisitiones , and her work of that period that has come down to us seems to touch only on rather superficial matters." Gray adds on to say "The inclination of sympathetic mathematicians to praise her work rather than to provide substantive criticism from which she might learn was crippling to her mathematical development." Yet Marilyn Bailey Ogilvie recognizes that "Sophie Germain's creativity manifested itself in pure and applied mathematics ... [she] provided imaginative and provocative solutions to several important problems", and, as Petrovich proposes, it may have been her very lack of training that gave her unique insights and approaches. Louis Bucciarelli and Nancy Dworsky, Germain's biographers, summarize as follows: "All

6556-724: Is of the form h 2 + n f 2 {\displaystyle h^{2}+nf^{2}} , then x + y {\displaystyle x+y} is also of that form. Gauss replied with a counterexample: 15 11 + 8 11 {\displaystyle 15^{11}+8^{11}} can be written as h 2 + 11 f 2 {\displaystyle h^{2}+11f^{2}} , but 15 + 8 {\displaystyle 15+8} cannot. Although Gauss thought well of Germain, his replies to her letters were often delayed, and he generally did not review her work. Eventually his interests turned away from number theory, and in 1809

6705-638: Is only survived by the Plimpton 322 tablet, some authors assert that Babylonian algebra was exceptionally well developed and included the foundations of modern elementary algebra . Late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt . In book nine of Euclid's Elements , propositions 21–34 are very probably influenced by Pythagorean teachings ; it

6854-608: Is the master of us all." Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." His 866 publications and his correspondence are being collected in the Opera Omnia Leonhard Euler which, when completed, will consist of 81 quartos . He spent most of his adult life in Saint Petersburg , Russia, and in Berlin , then

7003-457: Is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions , which are generalizations of the Riemann zeta function , a key analytic object at

7152-430: Is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian circuit . This solution is considered to be the first theorem of graph theory . Euler also discovered the formula V − E + F = 2 {\displaystyle V-E+F=2} relating the number of vertices, edges, and faces of

7301-425: Is very simple material ("odd times even is even", "if an odd number measures [= divides] an even number, then it also measures [= divides] half of it"), but it is all that is needed to prove that 2 {\displaystyle {\sqrt {2}}} is irrational . Pythagorean mystics gave great importance to the odd and the even. The discovery that 2 {\displaystyle {\sqrt {2}}}

7450-758: Is well known in analysis for his frequent use and development of power series , the expression of functions as sums of infinitely many terms, such as e x = ∑ n = 0 ∞ x n n ! = lim n → ∞ ( 1 0 ! + x 1 ! + x 2 2 ! + ⋯ + x n n ! ) . {\displaystyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}=\lim _{n\to \infty }\left({\frac {1}{0!}}+{\frac {x}{1!}}+{\frac {x^{2}}{2!}}+\cdots +{\frac {x^{n}}{n!}}\right).} Euler's use of power series enabled him to solve

7599-512: The Supplément to his second edition of the Théorie des Nombres , where he calls it très ingénieuse ("very ingenious"). See also Her work on Fermat's Last Theorem below. Germain's interest in number theory was renewed when she read Carl Friedrich Gauss 's monumental work Disquisitiones Arithmeticae . After three years of working through the exercises and trying her own proofs for some of

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7748-549: The Institutiones calculi differentialis was published. In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences and of the French Academy of Sciences . Notable students of Euler in Berlin included Stepan Rumovsky , later considered as the first Russian astronomer. In 1748 he declined an offer from the University of Basel to succeed the recently deceased Johann Bernoulli. In 1753 he bought

7897-1036: The Basel problem , finding the sum of the reciprocals of squares of every natural number, in 1735 (he provided a more elaborate argument in 1741). The Basel problem was originally posed by Pietro Mengoli in 1644, and by the 1730s was a famous open problem, popularized by Jacob Bernoulli and unsuccessfully attacked by many of the leading mathematicians of the time. Euler found that: ∑ n = 1 ∞ 1 n 2 = lim n → ∞ ( 1 1 2 + 1 2 2 + 1 3 2 + ⋯ + 1 n 2 ) = π 2 6 . {\displaystyle \sum _{n=1}^{\infty }{1 \over n^{2}}=\lim _{n\to \infty }\left({\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots +{\frac {1}{n^{2}}}\right)={\frac {\pi ^{2}}{6}}.} Euler introduced

8046-454: The Bernoulli numbers , Fourier series , Euler numbers , the constants e and π , continued fractions, and integrals. He integrated Leibniz 's differential calculus with Newton's Method of Fluxions , and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as

8195-475: The Eiffel Tower was built and the architects inscribed the names of 72 great French scientists, Germain's name was not among them, despite the salience of her work to the tower's construction. Mozans asked: "Was she excluded from this list ... because she was a woman? It would seem so." Concerning her early work in number theory, J. H. Sampson states: "She was clever with formal algebraic manipulations; but there

8344-477: The Euler approximations . The most notable of these approximations are Euler's method and the Euler–Maclaurin formula . Number theory Integers can be considered either in themselves or as solutions to equations ( Diophantine geometry ). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function ) that encode properties of

8493-540: The Imperial Russian Academy of Sciences in Saint Petersburg in 1725, leaving Euler with the assurance they would recommend him to a post when one was available. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia. When Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726, Euler eagerly accepted

8642-507: The St. Petersburg Academy , which had retained him as a member and paid him an annual stipend. Euler's Introductio in Analysin Infinitorum was published in two parts in 1748. In addition to his own research, Euler supervised the library, the observatory, the botanical garden, and the publication of calendars and maps from which the academy derived income. He was even involved in the design of

8791-466: The imaginary unit − 1 {\displaystyle {\sqrt {-1}}} , the Greek letter Σ {\displaystyle \Sigma } (capital sigma ) to express summations , the Greek letter Δ {\displaystyle \Delta } (capital delta ) for finite differences , and lowercase letters to represent the sides of a triangle while representing

8940-514: The largest known prime until 1867. Euler also contributed major developments to the theory of partitions of an integer . In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg . The city of Königsberg , Prussia was set on the Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem

9089-486: The Euler–Mascheroni constant, and studied its relationship with the harmonic series , the gamma function , and values of the Riemann zeta function . Euler introduced the use of the exponential function and logarithms in analytic proofs . He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers , thus greatly expanding

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9238-751: The Foundation Sophie Germain, is conferred by the Academy of Sciences in Paris. Its purpose is to honour a French mathematician for research in the foundations of mathematics . This award, in the amount of €8,000, was established in 2003, under the auspices of the Institut de France . Leonhard Euler Leonhard Euler ( / ˈ ɔɪ l ər / OY -lər ; German: [ˈleːɔnhaʁt ˈʔɔʏlɐ] , Swiss Standard German: [ˈleɔnhard ˈɔʏlər] ; 15 April 1707 – 18 September 1783)

9387-403: The academy beginning in 1720) for the first time. The problem posed that year was to find the best way to place the masts on a ship. Pierre Bouguer , who became known as "the father of naval architecture", won and Euler took second place. Over the years, Euler entered this competition 15 times, winning 12 of them. Johann Bernoulli's two sons, Daniel and Nicolaus , entered into service at

9536-458: The academy was still not fully satisfied. Germain had derived the correct differential equation (a special case of the Kirchhoff–Love equation ), but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler, which led to incorrect boundary conditions. Here is Germain's final equation for the vibration of a plane lamina: where N

9685-531: The age of eight, Euler was sent to live at his maternal grandmother's house and enrolled in the Latin school in Basel. In addition, he received private tutoring from Johannes Burckhardt, a young theologian with a keen interest in mathematics. In 1720, at thirteen years of age, Euler enrolled at the University of Basel . Attending university at such a young age was not unusual at the time. The course on elementary mathematics

9834-485: The angles as capital letters. He gave the current definition of the constant e {\displaystyle e} , the base of the natural logarithm , now known as Euler's number . Euler is also credited with being the first to develop graph theory (partly as a solution for the problem of the Seven Bridges of Königsberg , which is also considered the first practical application of topology). He also became famous for, among many other accomplishments, providing

9983-528: The ascension of the twelve-year-old Peter II . The nobility, suspicious of the academy's foreign scientists, cut funding for Euler and his colleagues and prevented the entrance of foreign and non-aristocratic students into the Gymnasium and universities. Conditions improved slightly after the death of Peter II in 1730 and the German-influenced Anna of Russia assumed power. Euler swiftly rose through

10132-474: The basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied quadratic forms in full generality (as opposed to m X 2 + n Y 2 {\displaystyle mX^{2}+nY^{2}} )—defining their equivalence relation, showing how to put them in reduced form, etc. Adrien-Marie Legendre (1752–1833)

10281-447: The biography "is inaccurate and the notes and bibliography are unreliable". Nevertheless, it quotes the mathematician Claude-Louis Navier as saying that "it is a work which few men are able to read and which only one woman was able to write". Germain's contemporaries also had good things to say relating to her work in mathematics. Gauss certainly thought highly of her and recognized that European culture presented special difficulties to

10430-625: The caliph Al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (the Sindhind , which may or may not be Brahmagupta's Brāhmasphuṭasiddhānta ). Diophantus's main work, the Arithmetica , was translated into Arabic by Qusta ibn Luqa (820–912). Part of the treatise al-Fakhri (by al-Karajī , 953 – ca. 1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary Ibn al-Haytham knew what would later be called Wilson's theorem . Other than

10579-417: The capital of Prussia . Euler is credited for popularizing the Greek letter π {\displaystyle \pi } (lowercase pi ) to denote the ratio of a circle's circumference to its diameter , as well as first using the notation f ( x ) {\displaystyle f(x)} for the value of a function, the letter i {\displaystyle i} to express

10728-472: The centenary of her life, a street and a girls' school were named after her. The Academy of Sciences established the Sophie Germain Prize in her honour. Marie-Sophie Germain was born in a house on Rue Saint-Denis on 1 April 1776, in Paris, France. According to most sources, her father, Ambroise-François, was a wealthy silk merchant, though some believe he was a goldsmith . In 1789, he was elected as

10877-451: The complex numbers C are an extension of the reals R , and the reals R are an extension of the rationals Q .) Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions—that is, extensions L of K such that the Galois group Gal( L / K ) of L over K is an abelian group —are relatively well understood. Their classification

11026-498: The constant γ = lim n → ∞ ( 1 + 1 2 + 1 3 + 1 4 + ⋯ + 1 n − ln ⁡ ( n ) ) ≈ 0.5772 , {\displaystyle \gamma =\lim _{n\rightarrow \infty }\left(1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\cdots +{\frac {1}{n}}-\ln(n)\right)\approx 0.5772,} now known as Euler's constant or

11175-559: The continuing turmoil in Russia, Euler left St. Petersburg in June 1741 to take up a post at the Berlin Academy , which he had been offered by Frederick the Great of Prussia . He lived for 25 years in Berlin , where he wrote several hundred articles. In 1748 his text on functions called the Introductio in analysin infinitorum was published and in 1755 a text on differential calculus called

11324-479: The damage caused to Euler's estate, with Empress Elizabeth of Russia later adding a further payment of 4000 rubles—an exorbitant amount at the time. Euler decided to leave Berlin in 1766 and return to Russia. During his Berlin years (1741–1766), Euler was at the peak of his productivity. He wrote 380 works, 275 of which were published. This included 125 memoirs in the Berlin Academy and over 100 memoirs sent to

11473-704: The death of Archimedes intrigued her. Germain thought that if the geometry method, which at that time referred to all of pure mathematics, could hold such fascination for Archimedes, it was a subject worthy of study. So she pored over every book on mathematics in her father's library, even teaching herself Latin and Greek, so she could read works like those of Sir Isaac Newton and Leonhard Euler . She also enjoyed Traité d'Arithmétique by Étienne Bézout and Le Calcul Différentiel by Jacques Antoine-Joseph Cousin . Later, Cousin visited Germain at home, encouraging her in her studies. Germain's parents did not at all approve of her sudden fascination with mathematics, which

11622-537: The development of the prime number theorem . Euler's interest in number theory can be traced to the influence of Christian Goldbach , his friend in the St. Petersburg Academy. Much of Euler's early work on number theory was based on the work of Pierre de Fermat . Euler developed some of Fermat's ideas and disproved some of his conjectures, such as his conjecture that all numbers of the form 2 2 n + 1 {\textstyle 2^{2^{n}}+1} ( Fermat numbers ) are prime. Euler linked

11771-500: The discovery of the laws of equilibrium and movement of elastic solids." On 27 June 1831, she died in the house at 13 rue de Savoie. Despite Germain's intellectual achievements, her death certificate lists her as a " rentière – annuitant " (property holder), not a " mathématicienne ". But her work was not unappreciated by everyone. When the matter of honorary degrees came up at the University of Göttingen in 1837—six years after Germain's death—Gauss lamented: "she [Germain] proved to

11920-496: The discussion of quadratic forms in Disquisitiones arithmeticae can be restated in terms of ideals and norms in quadratic fields. (A quadratic field consists of all numbers of the form a + b d {\displaystyle a+b{\sqrt {d}}} , where a {\displaystyle a} and b {\displaystyle b} are rational numbers and d {\displaystyle d}

12069-506: The educated world's response to the publication in 1821 of Germain's prize-winning essay "ranged from polite to indifferent". Yet, some critics had high praise for it. Of her essay in 1821, Cauchy said: "[it] was a work for which the name of its author and the importance of the subject both deserved the attention of mathematicians". Germain was also included in H. J. Mozans ' 1913 book Woman in Science , although Marilyn Bailey Ogilvie claims that

12218-428: The entirety of the poem, along with stating the first and last sentence on each page of the edition from which he had learnt it. Euler's eyesight worsened throughout his mathematical career. In 1738, three years after nearly expiring from fever, he became almost blind in his right eye. Euler blamed the cartography he performed for the St. Petersburg Academy for his condition, but the cause of his blindness remains

12367-407: The essay she pointed out some of the errors in his method. In 1826 she submitted a revised version of her 1821 essay to the academy. According to Andrea Del Centina, the revision included attempts to clarify her work by "introducing certain simplifying hypotheses". This put the academy in an awkward position, as they felt the paper to be "inadequate and trivial", but they did not want to "treat her as

12516-529: The evidence argues that Sophie Germain had a mathematical brilliance that never reached fruition due to a lack of rigorous training available only to men." Germain was referenced and quoted in David Auburn 's 2001 play Proof . The protagonist is a young struggling female mathematician, Catherine, who found great inspiration in the work of Germain. Germain was also mentioned in John Madden's film adaptation of

12665-503: The fall of 1811 and did not win the prize. The judging commission felt that "the true equations of the movement were not established", even though "the experiments presented ingenious results". Lagrange was able to use Germain's work to derive an equation that was "correct under special assumptions". The contest was extended by two years, and Germain decided to try again for the prize. At first Legendre continued to offer support, but then he refused all help. Germain's anonymous 1813 submission

12814-435: The first case of Fermat's Last Theorem for all odd primes less than 1700. In an unpublished manuscript titled Remarque sur l'impossibilité de satisfaire en nombres entiers a l'équation x + y = z , Germain showed that any counterexamples to Fermat's theorem for p  > 5 must be numbers "whose size frightens the imagination", around 40 digits long. Germain did not publish this work. Her theorem

12963-394: The first column reads: "The takiltum of the diagonal which has been subtracted such that the width..." The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity which is implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and then reordered by c /

13112-579: The following, commonly called " Sophie Germain's theorem ": Let p be an odd prime. If there exists an auxiliary prime P  = 2 Np  + 1 ( N is any positive integer not divisible by 3) such that: Then the first case of Fermat's Last Theorem holds true for p . Germain used this result to prove the first case of Fermat's Last Theorem for all odd primes p  < 100, but according to Andrea Del Centina, "she had actually shown that it holds for every exponent p  < 197". L. E. Dickson later used Germain's theorem to prove

13261-663: The force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci . My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry! However, the disappointment was almost surely unwarranted from a technical perspective. Euler's calculations look likely to be correct, even if Euler's interactions with Frederick and those constructing his fountain may have been dysfunctional. Throughout his stay in Berlin, Euler maintained

13410-555: The form p  = 8 k  + 7. However, her proof contained a weak assumption, and Gauss's reply did not comment on Germain's proof. Around 1807 (sources differ), during the Napoleonic wars, the French were occupying the German town of Braunschweig , where Gauss lived. Germain, concerned that he might suffer the fate of Archimedes, wrote to General Pernety ( Joseph Marie de Pernety ),

13559-557: The general public to mean " elementary calculations "; it has also acquired other meanings in mathematical logic , as in Peano arithmetic , and computer science , as in floating-point arithmetic .) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is commonly preferred as an adjective to number-theoretic . The earliest historical find of an arithmetical nature

13708-505: The greatest common divisor of two numbers (the Euclidean algorithm ; Elements , Prop. VII.2) and the first known proof of the infinitude of primes ( Elements , Prop. IX.20). In 1773, Lessing published an epigram he had found in a manuscript during his work as a librarian; it claimed to be a letter sent by Archimedes to Eratosthenes . The epigram proposed what has become known as Archimedes's cattle problem ; its solution (absent from

13857-414: The integer n that are coprime to n . Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem . He contributed significantly to the theory of perfect numbers , which had fascinated mathematicians since Euclid . He proved that the relationship shown between even perfect numbers and Mersenne primes (which he had earlier proved) was one-to-one,

14006-403: The integers, primes or other number-theoretic objects in some fashion ( analytic number theory ). One may also study real numbers in relation to rational numbers; for example, as approximated by the latter ( Diophantine approximation ). The older term for number theory is arithmetic . By the early twentieth century, it had been superseded by number theory . (The word arithmetic is used by

14155-873: The lack of unique factorisation in algebraic number fields. (For example, in the field generated by the rationals and − 5 {\displaystyle {\sqrt {-5}}} , the number 6 {\displaystyle 6} can be factorised both as 6 = 2 ⋅ 3 {\displaystyle 6=2\cdot 3} and 6 = ( 1 + − 5 ) ( 1 − − 5 ) {\displaystyle 6=(1+{\sqrt {-5}})(1-{\sqrt {-5}})} ; all of 2 {\displaystyle 2} , 3 {\displaystyle 3} , 1 + − 5 {\displaystyle 1+{\sqrt {-5}}} and 1 − − 5 {\displaystyle 1-{\sqrt {-5}}} are irreducible, and thus, in

14304-463: The left eye as well. However, his condition appeared to have little effect on his productivity. With the aid of his scribes, Euler's productivity in many areas of study increased; and, in 1775, he produced, on average, one mathematical paper every week. In St. Petersburg on 18 September 1783, after a lunch with his family, Euler was discussing the newly discovered planet Uranus and its orbit with Anders Johan Lexell when he collapsed and died from

14453-479: The letters ceased. Despite the friendship of Germain and Gauss, they never met. When Germain's correspondence with Gauss ceased, she took interest in a contest sponsored by the Paris Academy of Sciences concerning Ernst Chladni 's experiments with vibrating metal plates. The object of the competition, as stated by the academy, was "to give the mathematical theory of the vibration of an elastic surface and to compare

14602-405: The lines later developed fully by Gauss. In his old age, he was the first to prove Fermat's Last Theorem for n = 5 {\displaystyle n=5} (completing work by Peter Gustav Lejeune Dirichlet , and crediting both him and Sophie Germain ). In his Disquisitiones Arithmeticae (1798), Carl Friedrich Gauss (1777–1855) proved the law of quadratic reciprocity and developed

14751-505: The manuscript) requires solving an indeterminate quadratic equation (which reduces to what would later be misnamed Pell's equation ). As far as it is known, such equations were first successfully treated by the Indian school . It is not known whether Archimedes himself had a method of solution. Very little is known about Diophantus of Alexandria ; he probably lived in the third century AD, that is, about five hundred years after Euclid. Six out of

14900-490: The mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early Hellenistic period . In the case of number theory, this means, by and large, Plato and Euclid , respectively. While Asian mathematics influenced Greek and Hellenistic learning, it seems to be the case that Greek mathematics is also an indigenous tradition. Eusebius , PE X, chapter 4 mentions of Pythagoras : "In fact

15049-483: The modern notation for the trigonometric functions , the letter e for the base of the natural logarithm (now also known as Euler's number ), the Greek letter Σ for summations and the letter i to denote the imaginary unit . The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it originated with Welsh mathematician William Jones . The development of infinitesimal calculus

15198-426: The most important tools of analytic number theory are the circle method , sieve methods and L-functions (or, rather, the study of their properties). The theory of modular forms (and, more generally, automorphic forms ) also occupies an increasingly central place in the toolbox of analytic number theory. One may ask analytic questions about algebraic numbers , and use analytic means to answer such questions; it

15347-694: The most interesting questions in each area remain open and are being actively worked on. The term elementary generally denotes a method that does not use complex analysis . For example, the prime number theorem was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by Erdős and Selberg . The term is somewhat ambiguous: for example, proofs based on complex Tauberian theorems (for example, Wiener–Ikehara ) are often seen as quite enlightening but not elementary, in spite of using Fourier analysis , rather than complex analysis as such. Here as elsewhere, an elementary proof may be longer and more difficult for most readers than

15496-495: The motion of rigid bodies . He also made substantial contributions to the study of elastic deformations of solid objects. Leonhard Euler was born on 15 April 1707, in Basel to Paul III Euler, a pastor of the Reformed Church , and Marguerite (née Brucker), whose ancestors include a number of well-known scholars in the classics. He was the oldest of four children, having two younger sisters, Anna Maria and Maria Magdalena, and

15645-450: The nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges . In doing so, he discovered the connection between the Riemann zeta function and prime numbers; this is known as the Euler product formula for the Riemann zeta function . Euler invented the totient function φ( n ), the number of positive integers less than or equal to

15794-407: The new system of education made the "lecture notes available to all who asked". The new method also required the students to "submit written observations". Germain obtained the lecture notes and began sending her work to Joseph Louis Lagrange , a faculty member. She used the name of a former student Monsieur Antoine-Auguste Le Blanc, "fearing", as she later explained to Gauss, "the ridicule attached to

15943-541: The notion of a mathematical function . He is also known for his work in mechanics , fluid dynamics , optics , astronomy , and music theory . Euler is regarded as one of the greatest, most prolific mathematicians in history and the greatest of the 18th century. Several great mathematicians who produced their work after Euler's death have recognised his importance in the field as shown by quotes attributed to many of them: Pierre-Simon Laplace expressed Euler's influence on mathematics by stating, "Read Euler, read Euler, he

16092-598: The offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg in May 1727. He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian, settled into life in Saint Petersburg and took on an additional job as

16241-403: The polar opposite of Voltaire , who enjoyed a high place of prestige at Frederick's court. Euler was not a skilled debater and often made it a point to argue subjects that he knew little about, making him the frequent target of Voltaire's wit. Frederick also expressed disappointment with Euler's practical engineering abilities, stating: I wanted to have a water jet in my garden: Euler calculated

16390-592: The pseudonym of Monsieur LeBlanc). One of the pioneers of elasticity theory , she won the grand prize from the Paris Academy of Sciences for her essay on the subject. Her work on Fermat's Last Theorem provided a foundation for mathematicians exploring the subject for hundreds of years after. Because of prejudice against her sex, she was unable to make a career out of mathematics, but she worked independently throughout her life. Before her death, Gauss had recommended that she be awarded an honorary degree, but that never occurred. On 27 June 1831, she died from breast cancer. At

16539-514: The ranks in the academy and was made a professor of physics in 1731. He also left the Russian Navy, refusing a promotion to lieutenant . Two years later, Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. In January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell . Frederick II had made an attempt to recruit

16688-399: The rationals—the subjects of arithmetic), on the one hand, and lengths and proportions (which may be identified with real numbers, whether rational or not), on the other hand. The Pythagorean tradition spoke also of so-called polygonal or figurate numbers . While square numbers , cubic numbers , etc., are seen now as more natural than triangular numbers , pentagonal numbers , etc.,

16837-755: The roots of the subject. This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function. An algebraic number is any complex number that is a solution to some polynomial equation f ( x ) = 0 {\displaystyle f(x)=0} with rational coefficients; for example, every solution x {\displaystyle x} of x 5 + ( 11 / 2 ) x 3 − 7 x 2 + 9 = 0 {\displaystyle x^{5}+(11/2)x^{3}-7x^{2}+9=0} (say)

16986-401: The rudiments of analysis." Petrovich adds: "This proved to be a major handicap when she could no longer be regarded as a young prodigy to be admired but was judged by her peer mathematicians." Notwithstanding the problems with Germain's theory of vibrations, Gray states that "Germain's work was fundamental in the development of a general theory of elasticity." Mozans writes, however, that when

17135-520: The said Pythagoras, while busily studying the wisdom of each nation, visited Babylon, and Egypt, and all Persia, being instructed by the Magi and the priests: and in addition to these he is related to have studied under the Brahmans (these are Indian philosophers); and from some he gathered astrology, from others geometry, and arithmetic and music from others, and different things from different nations, and only from

17284-591: The same name in a conversation between Catherine (Gwyneth Paltrow) and Hal (Jake Gyllenhaal). In the fictional work " The Last Theorem " by Arthur C. Clarke and Frederik Pohl , Sophie Germain was credited with inspiring the central character, Ranjit Subramanian, to solve Fermat's Last Theorem . A musical about Sophie Germain's life, entitled The Limit , premiered at VAULT Festival in London, 2019. The Sophie Germain Prize (French: Prix Sophie Germain ), awarded annually by

17433-504: The scope of mathematical applications of logarithms. He also defined the exponential function for complex numbers and discovered its relation to the trigonometric functions . For any real number φ (taken to be radians), Euler's formula states that the complex exponential function satisfies e i φ = cos ⁡ φ + i sin ⁡ φ {\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi } which

17582-559: The second rather than the first definition: some of sieve theory, for instance, uses little analysis, yet it does belong to analytic number theory. The following are examples of problems in analytic number theory: the prime number theorem , the Goldbach conjecture (or the twin prime conjecture , or the Hardy–Littlewood conjectures ), the Waring problem and the Riemann hypothesis . Some of

17731-661: The services of Euler for his newly established Berlin Academy in 1740, but Euler initially preferred to stay in St Petersburg. But after Empress Anna died and Frederick II agreed to pay 1600 ecus (the same as Euler earned in Russia) he agreed to move to Berlin. In 1741, he requested permission to leave to Berlin, arguing he was in need of a milder climate for his eyesight. The Russian academy gave its consent and would pay him 200 rubles per year as one of its active members. Concerned about

17880-512: The study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period (17th to early 19th centuries). The Chinese remainder theorem appears as an exercise in Sunzi Suanjing (3rd, 4th or 5th century CE). (There is one important step glossed over in Sunzi's solution: it is the problem that was later solved by Āryabhaṭa 's Kuṭṭaka – see below .) The result

18029-427: The subject and, as a judge on the academy commission, had had access to her work). Germain submitted her third paper, " Recherches sur la théorie des surfaces élastiques ", under her own name, and on 8 January 1816 she became the first woman to win a prize from the Paris Academy of Sciences. She did not appear at the ceremony to receive her award. Although Germain had at last been awarded the prix extraordinaire ,

18178-475: The subject of speculation. Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as " Cyclops ". Euler remarked on his loss of vision, stating "Now I will have fewer distractions." In 1766 a cataract in his left eye was discovered. Though couching of the cataract temporarily improved his vision, complications ultimately rendered him almost totally blind in

18327-399: The subject. This has been called the "rebirth" of modern number theory, after Fermat's relative lack of success in getting his contemporaries' attention for the subject. Euler's work on number theory includes the following: Joseph-Louis Lagrange (1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations—for instance, the four-square theorem and

18476-452: The systematic study of indefinite quadratic equations—in particular, the misnamed Pell equation , in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general procedure (the chakravala , or "cyclic method") for solving Pell's equation

18625-409: The theorems, she wrote, again under the pseudonym of M. Le Blanc, to the author himself, who was one year younger than she. The first letter, dated 21 November 1804, discussed Gauss's Disquisitiones and presented some of Germain's work on Fermat's Last Theorem . In the letter, Germain claimed to have proved the theorem for n  =  p  − 1, where p is a prime number of

18774-518: The theory of quadratic forms (in particular, defining their composition). He also introduced some basic notation ( congruences ) and devoted a section to computational matters, including primality tests. The last section of the Disquisitiones established a link between roots of unity and number theory: The theory of the division of the circle...which is treated in sec. 7 does not belong by itself to arithmetic, but its principles can only be drawn from higher arithmetic. In this way, Gauss arguably made

18923-496: The theory to experimental evidence". Lagrange's comment that a solution to the problem would require the invention of a new branch of analysis deterred all but two contestants, Denis Poisson and Germain. Then Poisson was elected to the academy, thus becoming a judge instead of a contestant, and leaving Germain as the only entrant to the competition. In 1809 Germain began work. Legendre assisted by giving her equations, references, and current research. She submitted her paper early in

19072-540: The thirteen books of Diophantus's Arithmetica survive in the original Greek and four more survive in an Arabic translation. The Arithmetica is a collection of worked-out problems where the task is invariably to find rational solutions to a system of polynomial equations, usually of the form f ( x , y ) = z 2 {\displaystyle f(x,y)=z^{2}} or f ( x , y , z ) = w 2 {\displaystyle f(x,y,z)=w^{2}} . Thus, nowadays,

19221-461: The water fountains at Sanssouci , the King's summer palace. The political situation in Russia stabilized after Catherine the Great's accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Petersburg Academy. His conditions were quite exorbitant—a 3000 ruble annual salary, a pension for his wife, and the promise of high-ranking appointments for his sons. At the university he

19370-572: The wise men of Greece did he get nothing, wedded as they were to a poverty and dearth of wisdom: so on the contrary he himself became the author of instruction to the Greeks in the learning which he had procured from abroad." Aristotle claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans, and Cicero repeats this claim: Platonem ferunt didicisse Pythagorea omnia ("They say Plato learned all things Pythagorean"). Plato had

19519-423: The work of Bernhard Riemann (1859) on the zeta function is the canonical starting point; Jacobi's four-square theorem (1839), which predates it, belongs to an initially different strand that has by now taken a leading role in analytic number theory ( modular forms ). The history of each subfield is briefly addressed in its own section below; see the main article of each subfield for fuller treatments. Many of

19668-503: The world that even a woman can accomplish something worthwhile in the most rigorous and abstract of the sciences and for that reason would well have deserved an honorary degree". Germain's resting place in the Père Lachaise Cemetery in Paris is marked by a gravestone. At the centennial celebration of her life, a street and a girls' school were named after her, and a plaque was placed at the house where she died. The school houses

19817-498: Was Johann Albrecht Euler , whose godfather was Christian Goldbach . Three years after his wife's death in 1773, Euler married her half-sister, Salome Abigail Gsell (1723–1794). This marriage lasted until his death in 1783. His brother Johann Heinrich settled in St. Petersburg in 1735 and was employed as a painter at the academy. Early in his life, Euler memorized the entirety of the Aeneid by Virgil , and by old age, could recite

19966-405: Was a Swiss mathematician , physicist , astronomer , geographer , logician , and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory , complex analysis , and infinitesimal calculus . He introduced much of modern mathematical terminology and notation , including

20115-604: Was also named Marie-Madeline, and this plethora of "Maries" may have been the reason she went by Sophie. Germain's nephew Armand-Jacques Lherbette, Marie-Madeline's son, published some of Germain's work after she died (see Work in Philosophy ). When Germain was 13, the Bastille fell , and the revolutionary atmosphere of the city forced her to stay inside. For entertainment, she turned to her father's library. Here she found J. E. Montucla's L'Histoire des Mathématiques , and his story of

20264-539: Was assisted by his student Anders Johan Lexell . While living in St. Petersburg, a fire in 1771 destroyed his home. On 7 January 1734, he married Katharina Gsell (1707–1773), daughter of Georg Gsell , a painter from the Academy Gymnasium in Saint Petersburg. The young couple bought a house by the Neva River . Of their thirteen children, only five survived childhood, three sons and two daughters. Their first son

20413-519: Was at the forefront of 18th-century mathematical research, and the Bernoullis —family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour (in particular his reliance on the principle of the generality of algebra ), his ideas led to many great advances. Euler

20562-772: Was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island . In 1837, the Russian Academy of Sciences installed a new monument, replacing his overgrown grave plaque. To commemorate the 250th anniversary of Euler's birth in 1957, his tomb was moved to the Lazarevskoe Cemetery at the Alexander Nevsky Monastery . Euler worked in almost all areas of mathematics, including geometry , infinitesimal calculus , trigonometry , algebra , and number theory , as well as continuum physics , lunar theory , and other areas of physics . He

20711-427: Was called "the most remarkable formula in mathematics" by Richard Feynman . A special case of the above formula is known as Euler's identity , e i π + 1 = 0 {\displaystyle e^{i\pi }+1=0} Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations . He found

20860-452: Was finally found by Jayadeva (cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in Bhāskara II 's Bīja-gaṇita (twelfth century). Indian mathematics remained largely unknown in Europe until the late eighteenth century; Brahmagupta and Bhāskara's work was translated into English in 1817 by Henry Colebrooke . In the early ninth century,

21009-423: Was given by Johann Bernoulli , the younger brother of the deceased Jacob Bernoulli (who had taught Euler's father). Johann Bernoulli and Euler soon got to know each other better. Euler described Bernoulli in his autobiography: It was during this time that Euler, backed by Bernoulli, obtained his father's consent to become a mathematician instead of a pastor. In 1723, Euler received a Master of Philosophy with

21158-553: Was in her mind all the time she was studying elasticity. She outlined a strategy for a general proof of Fermat's Last Theorem, including a proof for a special case. Germain's letter to Gauss contained her substantial progress toward a proof. She asked Gauss whether her approach to the theorem was worth pursuing. Gauss never answered. Fermat's Last Theorem can be divided into two cases. Case 1 involves all powers p that do not divide any of x , y , or z . Case 2 includes all p that divide at least one of x , y , or z . Germain proposed

21307-418: Was in number theory, and her most significant contribution to number theory dealt with Fermat's Last Theorem. In 1815, after the elasticity contest, the academy offered a prize for a proof of Fermat's Last Theorem. It reawakened Germain's interest in number theory, and she wrote to Gauss again after ten years of no correspondence. In the letter, Germain said that number theory was her preferred field and that it

21456-565: Was later generalized with a complete solution called Da-yan-shu ( 大衍術 ) in Qin Jiushao 's 1247 Mathematical Treatise in Nine Sections which was translated into English in early 19th century by British missionary Alexander Wylie . There is also some numerical mysticism in Chinese mathematics, but, unlike that of the Pythagoreans, it seems to have led nowhere. Aside from a few fragments,

21605-457: Was still littered with mathematical errors, especially involving double integrals , and it received only an honorable mention because "the fundamental base of the theory [of elastic surfaces] was not established". The contest was extended once more, and Germain began work on her third attempt. This time she consulted with Poisson. In 1814 he published his own work on elasticity and did not acknowledge Germain's help (although he had worked with her on

21754-402: Was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the prime number theorem and Dirichlet's theorem on arithmetic progressions . He gave a full treatment of the equation a x 2 + b y 2 + c z 2 = 0 {\displaystyle ax^{2}+by^{2}+cz^{2}=0} and worked on quadratic forms along

21903-524: Was the object of the programme of class field theory , which was initiated in the late 19th century (partly by Kronecker and Eisenstein ) and carried out largely in 1900–1950. An example of an active area of research in algebraic number theory is Iwasawa theory . The Langlands program , one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields. The central problem of Diophantine geometry

22052-420: Was then thought inappropriate for a woman. When night came, they would deny her warm clothes and a fire for her bedroom to try to keep her from studying, but after they left, she would take out candles, wrap herself in quilts and do mathematics. After some time, her mother even secretly supported her. In 1794, when Germain was 18, the École Polytechnique opened. As a woman, Germain was barred from attending, but

22201-532: Was translated into multiple languages, published across Europe and in the United States, and became more widely read than any of his mathematical works. The popularity of the Letters testifies to Euler's ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist. Despite Euler's immense contribution to the academy's prestige and having been put forward as

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