Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.

Diophantus of Alexandria (Ancient Greek: Διόφαντος ὁ Ἀλεξανδρεύς. b. between A.D. 200 and 214, d. between 284 and 298 at age 84), sometimes called "the father of algebra", was an Alexandrian Greek mathematician^[1][2][3][4] and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost. In studying Arithmetica, Pierre de Fermat concluded that a certain equation considered by Diophantus had no solutions, and noted without elaboration that he had found "a truly marvelous proof of this proposition," now referred to as Fermat's Last Theorem. This led to tremendous advances in number theory, and the study of Diophantine equations ("Diophantine geometry") and of Diophantine approximations remain important areas of mathematical research. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation.

Biography[link]

Little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between A.D. 200 and 214 to 284 or 298. Much of our knowledge of the life of Diophantus is derived from a 5th century Greek anthology of number games and strategy puzzles. One of the problems (sometimes called his epitaph) states:

Hunc Diophantus habet tumulum qui tempora vitae illius, mira denotat arte tibi. Egit sex tantem juvenie; lanugine malas vestire hinc coepit parte duodecima. Septante uxori post haec sociatur, et anno formosus quinto nascitur inde puer. Semissem aetatis postquam attigit ille paternae, infelix subita morte peremptus obit. Quator aestater genitor lugere superstes cogitur, hinc annos illius assequere.

'Here lies Diophantus,' the wonder behold.

Through art algebraic, the stone tells how old:

'God gave him his boyhood one-sixth of his life,

One twelfth more as youth while whiskers grew rife;

And then yet one-seventh ere marriage begun;

In five years there came a bouncing new son.

Alas, the dear child of master and sage

After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.'

This puzzle implies that Diophantus lived to be 84 years old. However, the accuracy of the information cannot be independently confirmed.

In popular culture, this puzzle was the Puzzle No.142 in Professor Layton and Pandora's Box as one of the hardest solving puzzles in the game, which needed to be unlocked by solving other puzzles first.

Arithmetica[link]

The Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations. Of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arab books discovered in 1968 are also by Diophantus.^[5] Some Diophantine problems from Arithmetica have been found in Arabic sources.

It should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus.

“Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems. For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos’s solutions” ^{[6][dubious – discuss]}

History[link]

Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the so-called Dark Ages, since the study of ancient Greek had greatly declined. The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the early modern world, copied by, and thus known to, medieval Byzantine scholars. In addition, some portion of the Arithmetica probably survived in the Arab tradition (see above). In 1463 German mathematician Regiomontanus wrote:

“No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hidden . . . .”

Arithmetica was first translated from Greek into Latin by Bombelli in 1570, but the translation was never published. However, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander. The best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins.

Margin writing by Fermat and Chortasmenos[link]

Problem II.8 in the Arithmetica (edition of 1670), annotated with Fermat's comment which became Fermat's Last Theorem.

The 1621 edition of Arithmetica by Bachet gained fame after Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy:

“If an integer n is greater than 2, then Failed to parse (Missing texvc executable; please see math/README to configure.): a^n + b^n = c^n

has no solutions in non-zero integers a, b, and c. I have a truly marvelous proof of this proposition which this margin is too narrow to contain.”

Fermat's proof was never found, and the problem of finding a proof for the theorem went unsolved for centuries. A proof was finally found in 1994 by Andrew Wiles after working on it for seven years. It is believed that Fermat did not actually have the proof he claimed to have. Although the original copy in which Fermat wrote this is lost today, Fermat's son edited the next edition of Diophantus, published in 1670. Even though the text is otherwise inferior to the 1621 edition, Fermat's annotations—including the "Last Theorem"—were printed in this version.

Fermat was not the first mathematician so moved to write in his own marginal notes to Diophantus; the Byzantine scholar John Chortasmenos (14th/15th C.) had written "Thy soul, Diophantus, be with Satan because of the difficulty of your theorems" next to the same problem.

Other works[link]

Diophantus wrote several other books besides Arithmetica, but very few of them have survived.

[edit] The Porisms

Diophantus himself refers to a work which consists of a collection of lemmas called The Porisms (or Porismata), but this book is entirely lost. Some scholars think that The porisms may have actually been a section of Arithmetica that is now lost.^{[citation needed]}

Although The Porisms is lost, we know three lemmas contained there, since Diophantus refers to them in the Arithmetica. One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i.e. given any a and b, with a > b, there exist c and d, all positive and rational, such that

Failed to parse (Missing texvc executable; please see math/README to configure.): a^3 - b^3 = c^3 + d^3.\

Polygonal numbers and geometric elements[link]

Diophantus is also known to have written on polygonal numbers, a topic of great interest to Pythagoras and Pythagoreans. Fragments of a book dealing with polygonal numbers are extant.

A book called Preliminaries to the Geometric Elements has been traditionally attributed to Hero of Alexandria. It has been studied recently by Wilbur Knorr, who suggested that the attribution to Hero is incorrect, and that the true author is Diophantus.^[7]

Influence[link]

Diophantus' work has had a large influence in history. Editions of Arithmetica exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the seventeenth and eighteenth centuries. Diophantus and his works have also influenced Arab mathematics and were of great fame among Arab mathematicians. Diophantus' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra. As far as we know Diophantus did not affect the lands of the Orient much and how much he affected India is a matter of debate.

The father of algebra?[link]

Diophantus is often called “the father of algebra" because he contributed greatly to number theory, mathematical notation, and because Arithmetica contains the earliest known use of syncopated notation.^[8] However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to Babylonian mathematics. For this, and other, reasons mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.”^[9]

Diophantine analysis[link]

Today Diophantine analysis is the area of study where integer (whole number) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in Arithmetica lead to quadratic equations. Diophantus looked at 3 different types of quadratic equations: Failed to parse (Missing texvc executable; please see math/README to configure.): ax^2 + bx = c , Failed to parse (Missing texvc executable; please see math/README to configure.): ax^2 = bx + c , and Failed to parse (Missing texvc executable; please see math/README to configure.): ax^2 + c = bx . The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers Failed to parse (Missing texvc executable; please see math/README to configure.): a, b, c

to all be positive in each of the three cases above.  Diophantus was always satisfied with a rational solution and did not require a whole number which means he accepted fractions as solutions to his problems. Diophantus considered negative or irrational square root solutions "useless", "meaningless", and even "absurd". To give one specific example, he calls the equation Failed to parse (Missing texvc executable; please see math/README to configure.): 4 = 4x + 20
'absurd' because it would lead to a negative value for x. One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations.

Mathematical notation[link]

Diophantus made important advances in mathematical notation. He was the first person to use algebraic notation and symbolism. Before him everyone wrote out equations completely. Diophantus introduced an algebraic symbolism that used an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. Mathematical historian Kurt Vogel states:

“The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... Since an abbreviation is also employed for the word ‘equals’, Diophantus took a fundamental step from verbal algebra towards symbolic algebra.”^{[citation needed]}

Although Diophantus made important advances in symbolism, he still lacked the necessary notation to express more general methods. This caused his work to be more concerned with particular problems rather than general situations. Some of the limitations of Diophantus' notation are that he only had notation for one unknown and, when problems involved more than a single unknown, Diophantus was reduced to expressing "first unknown", "second unknown", etc. in words. He also lacked a symbol for a general number n. Where we would write Failed to parse (Missing texvc executable; please see math/README to configure.): (12 + 6n)/(n^2 -3) , Diophantus has to resort to constructions like : ... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three.

Algebra still had a long way to go before very general problems could be written down and solved succinctly.

See also[link]

• Erdős–Diophantine graph
• Diophantus II.VIII
• Polynomial Diophantine equation

Notes[link]

References[link]

• Allard, A. "pes scolies aux arithmétiques de Diophante d'Alexandrie dans le Matritensis Bibo. Nat. 4678 et les Vaticani gr. 191 et 304," Byzantion 53. Brussels, 1983: 682-710.
• Christianidis, J. "Maxime Planude sur le sens du terme diophantien "plasmatikon"", Historia Scientiarum, 6 (1996)37-41.
• Christianidis, J. "Une interpretation byzantine de Diophante", Historia Mathematica, 25 (1998) 22-28.
• Heath, Sir Thomas, Diophantos of Alexandria: A Study in the History of Greek Algebra, Cambridge: Cambridge University Press, 1885, 1910.
• Robinson, D. C. and Luke Hodgkin. History of Mathematics, King's College London, 2003.
• Sesiano, Jacques. Books IV to VII of Diophantus’ Arithmetica in the Arabic translation attributed to Qusṭā ibn Lūqā, Heidelberg: Springer-Verlag, 1982. ISBN 0-387-90690-8.
• Tannery, P. L. Diophanti Alexandrini Opera omnia: cum Graecis commentariis, Lipsiae: In aedibus B.G. Teubneri, 1893-1895.
• Ver Eecke, P. Diophante d’Alexandrie: Les Six Livres Arithmétiques et le Livre des Nombres Polygones, Bruges: Desclée, De Brouwer, 1921.

Further reading[link]

• Heath, Sir Thomas (1981). A history of Greek mathematics. 2. Cambridge University Press: Cambridge. http://www.archive.org/details/ahistorygreekma00heatgoog. 
• Vogel, Kurt (1970). "Diophantus of Alexandria". Dictionary of Scientific Biography. 4. New York: Scribner. 

External links[link]

• O'Connor, John J.; Robertson, Edmund F., "Diophantus", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Diophantus.html .
• Diophantus's Riddle Diophantus' epitaph, by E. Weisstein
• Norbert Schappacher (2005). Diophantus of Alexandria : a Text and its History.
• Tannery's edition of the Works of Diophantus, now in the public domain (Classical Greek)
• Review of Sesiano's Diophantus Review of J. Sesiano, Books IV to VII of Diophantus' Arithmetica, by Jan P. Hogendijk
• Latin translation from 1575 by Wilhelm Xylander

Pierre de Fermat
Pierre de Fermat
Born	(1601-08-17)August 17, 1601 Beaumont-de-Lomagne, France
Died	January 12, 1665(1665-01-12) (aged 63) Castres, France
Residence	France
Nationality	French
Fields	Mathematics and Law
Known for	Number theory Analytic geometry Fermat's principle Probability Fermat's Last Theorem
Influences	François Viète

Pierre de Fermat (French pronunciation: [pjɛːʁ dəfɛʁˈma]; 17^[1] August 1601 or 1607/8^[2] – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the then unknown differential calculus, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for Fermat's Last Theorem, which he described in a note at the margin of a copy of Diophantus' Arithmetica.

Life and work[link]

Fermat was born in Beaumont-de-Lomagne, Tarn-et-Garonne, France; the late 15th century mansion where Fermat was born is now a museum. He was of Basque origin. Fermat's father was a wealthy leather merchant and second consul of Beaumont-de-Lomagne. Pierre had a brother and two sisters and was almost certainly brought up in the town of his birth. There is little evidence concerning his school education, but it may have been at the local Franciscan monastery.

Bust in the Salle des Illustres in Capitole de Toulouse

He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius's De Locis Planis to one of the mathematicians there. Certainly in Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima which he gave to Étienne d'Espagnet who clearly shared mathematical interests with Fermat. There he became much influenced by the work of François Viète.

From Bordeaux, Fermat went to Orléans where he studied law at the University. He received a degree in civil law before, in 1631, receiving the title of councillor at the High Court of Judicature in Toulouse, which he held for the rest of his life. Due to the office he now held he became entitled to change his name from Pierre Fermat to Pierre de Fermat. Fluent in Latin, Basque^{[citation needed]}, classical Greek, Italian, and Spanish, Fermat was praised for his written verse in several languages, and his advice was eagerly sought regarding the emendation of Greek texts.

He communicated most of his work in letters to friends, often with little or no proof of his theorems. This allowed him to preserve his status as an "amateur" while gaining the recognition he desired. This naturally led to priority disputes with contemporaries such as Descartes and Wallis. He developed a close relationship with Blaise Pascal^{[citation needed]}.^[3]

Anders Hald writes that, "The basis of Fermat's mathematics was the classical Greek treatises combined with Vieta's new algebraic methods."^[4]

Work[link]

Fermat's pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes' famous La géométrie. This manuscript was published posthumously in 1679 in "Varia opera mathematica", as Ad Locos Planos et Solidos Isagoge, ("Introduction to Plane and Solid Loci").^[5]

In Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum, Fermat developed a method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation.^[6] In these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature.

Pierre de Fermat

Fermat was the first person known to have evaluated the integral of general power functions. Using an ingenious trick, he was able to reduce this evaluation to the sum of geometric series.^[7] The resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus.^{[citation needed]}

In number theory, Fermat studied Pell's equation, perfect numbers, amicable numbers and what would later become Fermat numbers. It was while researching perfect numbers that he discovered the little theorem. He invented a factorization method—Fermat's factorization method—as well as the proof technique of infinite descent, which he used to prove Fermat's Last Theorem for the case n = 4. Fermat developed the two-square theorem, and the polygonal number theorem, which states that each number is a sum of three triangular numbers, four square numbers, five pentagonal numbers, and so on.

Although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss, doubted several of his claims, especially given the difficulty of some of the problems and the limited mathematical tools available to Fermat. His famous Last Theorem was first discovered by his son in the margin on his father's copy of an edition of Diophantus, and included the statement that the margin was too small to include the proof. He had not bothered to inform even Marin Mersenne of it. It was not proved until 1994, using techniques unavailable to Fermat.

Although he carefully studied, and drew inspiration from Diophantus, Fermat began a different tradition. Diophantus was content to find a single solution to his equations, even if it were an undesired fractional one. Fermat was interested only in integer solutions to his Diophantine equations, and he looked for all possible general solutions. He often proved that certain equations had no solution, which usually baffled his contemporaries.

Through his correspondence with Pascal in 1654, Fermat and Pascal helped lay the fundamental groundwork for the theory of probability. From this brief but productive collaboration on the problem of points, they are now regarded as joint founders of probability theory.^[8] Fermat is credited with carrying out the first ever rigorous probability calculation. In it, he was asked by a professional gambler why if he bet on rolling at least one six in four throws of a die he won in the long term, whereas betting on throwing at least one double-six in 24 throws of two dice resulted in him losing. Fermat subsequently proved why this was the case mathematically.^[9]

Fermat's principle of least time (which he used to derive Snell's law in 1657) was the first variational principle^[10] enunciated in physics since Hero of Alexandria described a principle of least distance in the first century CE. In this way, Fermat is recognized as a key figure in the historical development of the fundamental principle of least action in physics. The terms Fermat's principle and Fermat functional were named in recognition of this role.^[11]

Death[link]

Place of burial of Pierre de Fermat in Place Jean Jaurés, Castres, France. Translation of the plaque: in this place was buried on January 13, 1665, Pierre de Fermat, councilor of the chamber of Edit and mathematician of great renown, celebrated for his theorem (sic),
aⁿ + bⁿ ≠ cⁿ for n>2

He died at Castres, Tarn.^[2] The oldest and most prestigious high school in Toulouse is named after him: the Lycée Pierre de Fermat. French sculptor Théophile Barrau made a marble statue named Hommage à Pierre Fermat as tribute to Fermat, now at the Capitole of Toulouse.

Assessment of his work[link]

Holographic will handwritten by Fermat on 4 March 1660 — kept at the Departmental Archives of Haute-Garonne, in Toulouse

Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century. According to Peter L. Bernstein, in his book Against the Gods, Fermat "was a mathematician of rare power. He was an independent inventor of analytic geometry, he contributed to the early development of calculus, he did research on the weight of the earth, and he worked on light refraction and optics. In the course of what turned out to be an extended correspondence with Pascal, he made a significant contribution to the theory of probability. But Fermat's crowning achievement was in the theory of numbers."^[12]

Regarding Fermat's work in analysis, Isaac Newton wrote that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."^[13]

Of Fermat's number theoretic work, the great 20th-century mathematician André Weil wrote that "... what we possess of his methods for dealing with curves of genus 1 is remarkably coherent; it is still the foundation for the modern theory of such curves. It naturally falls into two parts; the first one ... may conveniently be termed a method of ascent, in contrast with the descent which is rightly regarded as Fermat's own."^[14] Regarding Fermat's use of ascent, Weil continued "The novelty consisted in the vastly extended use which Fermat made of it, giving him at least a partial equivalent of what we would obtain by the systematic use of the group theoretical properties of the rational points on a standard cubic."^[15] With his gift for number relations and his ability to find proofs for many of his theorems, Fermat essentially created the modern theory of numbers.

See also[link]

Notes[link]

Books referenced[link]

• Weil, André (1984). Number Theory: An approach through history From Hammurapi to Legendre. Birkhäuser. ISBN 0-8176-3141-0. 

Further reading[link]

• Mahoney, Michael Sean (1994). The mathematical career of Pierre de Fermat, 1601 - 1665. Princeton Univ. Press. ISBN 0-691-03666-7. 
• Singh, Simon (2002). Fermat's Last Theorem. Fourth Estate Ltd. ISBN 1-84115-791-0. 

External links[link]

Wikimedia Commons has media related to: Pierre de Fermat

• Fermat's Achievements
• Fermat's Fallibility at MathPages
• History of Fermat's Last Theorem (French)
• The Life and times of Pierre de Fermat (1601 - 1665) from W. W. Rouse Ball's History of Mathematics
• The Mathematics of Fermat's Last Theorem
• O'Connor, John J.; Robertson, Edmund F., "Pierre de Fermat", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Fermat.html .