Diophantus

Diophantus of Alexandria (Ancient Greek: Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 201 and 215; died around 84 years old, probably sometime between AD 285 and 299) was an Alexandrian Hellenistic mathematician, who was the author of a series of books called Arithmetica, many of which are now lost. Sometimes called "the father of algebra", his texts deal with solving algebraic equations. While reading Claude Gaspard Bachet de Méziriac's edition of Diophantus' Arithmetica, Pierre de Fermat concluded that a certain equation considered by Diophantus had no solutions, and noted in the margin 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 coined the term παρισότης (parisotes) to refer to an approximate equality.[1] This term was rendered as adaequalitas in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. 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-cover
Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.

Biography

Little is known about the life of Diophantus. He lived in Alexandria, Egypt, during the Roman era, probably from between AD 200 and 214 to 284 or 298. Diophantus has variously been described by historians as either Greek,[2][3][4] non-Greek,[5] Hellenized Egyptian,[6] Hellenized Babylonian,[7] Jewish, or Chaldean.[8] Much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the problems (sometimes called his epitaph) states:

'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' age x can be expressed as

x = x/6 + x/12 + x/7 + 5 + x/2 + 4

which gives x a value of 84 years. 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

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 Arabic books discovered in 1968 are also by Diophantus.[9] 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”.[10]

History

Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the so-called Dark Ages, since the study of ancient Greek, and literacy in general, 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. Scholia on Diophantus by the Byzantine Greek scholar John Chortasmenos (1370–1437) are preserved together with a comprehensive commentary written by the earlier Greek scholar Maximos Planudes (1260 – 1305), who produced an edition of Diophantus within the library of the Chora Monastery in Byzantine Constantinople.[11] 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

Diophantus-II-8-Fermat
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 an + bn = cn 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 (1370–1437) had written "Thy soul, Diophantus, be with Satan because of the difficulty of your other theorems and particularly of the present theorem" next to the same problem.[11]

Other works

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

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.

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

a3b3 = c3 + d3.

Polygonal numbers and geometric elements

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[12].

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.[13]

Influence

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 17th and 18th 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.

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.[14]

Diophantine analysis

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: ax2 + bx = c, ax2 = bx + c, and ax2 + 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 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 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

Diophantus made important advances in mathematical notation, becoming the first person known 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:[15]

“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.”

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 12 + 6n/n2 − 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

Notes

  1. ^ Katz, Mikhail G.; Schaps, David; Shnider, Steve (2013), "Almost Equal: The Method of Adequality from Diophantus to Fermat and Beyond", Perspectives on Science, 21 (3), arXiv:1210.7750, Bibcode:2012arXiv1210.7750K
  2. ^ Research Machines plc. (2004). The Hutchinson dictionary of scientific biography. Abingdon, Oxon: Helicon Publishing. p. 312. Diophantus (lived c. A.D. 270-280) Greek mathematician who, in solving linear mathematical problems, developed an early form of algebra.
  3. ^ Boyer, Carl B. (1991). "Revival and Decline of Greek Mathematics". A History of Mathematics (Second ed.). John Wiley & Sons, Inc. p. 178. ISBN 0-471-54397-7. At the beginning of this period, also known as the Later Alexandrian Age, we find the leading Greek algebraist, Diophantus of Alexandria, and toward its close there appeared the last significant Greek geometer, Pappus of Alexandria.
  4. ^ Cooke, Roger (1997). "The Nature of Mathematics". The History of Mathematics: A Brief Course. Wiley-Interscience. p. 7. ISBN 0-471-18082-3. Some enlargement in the sphere in which symbols were used occurred in the writings of the third-century Greek mathematician Diophantus of Alexandria, but the same defect was present as in the case of Akkadians.
  5. ^ H. Hankel (1874, 2nd ed. 1965), Zur Geschichte der Mathematik im Altertum und Mittelalter, Leipzig:

    Here, in the midst of this sad and barren landscape of the Greek accomplishments in arithmetic, suddenly springs up a man with youthful energy: Diophantus. Where does he come from, where does he go to? Who were his predecessors, who his successors? We do not know. It is all one big riddle. He lived in Alexandria. If a conjecture were permitted, I would say he was not Greek; ... if his writings were not in Greek, no-one would ever think that they were an outgrowth of Greek culture...

  6. ^ Victor J. Katz (1998). A History of Mathematics: An Introduction, p. 184. Addison Wesley, ISBN 0-321-01618-1.

    But what we really want to know is to what extent the Alexandrian mathematicians of the period from the first to the fifth centuries C.E. were Greek. Certainly, all of them wrote in Greek and were part of the Greek intellectual community of Alexandria. And most modern studies conclude that the Greek community coexisted [...] So should we assume that Ptolemy and Diophantus, Pappus and Hypatia were ethnically Greek, that their ancestors had come from Greece at some point in the past but had remained effectively isolated from the Egyptians? It is, of course, impossible to answer this question definitively. But research in papyri dating from the early centuries of the common era demonstrates that a significant amount of intermarriage took place between the Greek and Egyptian communities [...] And it is known that Greek marriage contracts increasingly came to resemble Egyptian ones. In addition, even from the founding of Alexandria, small numbers of Egyptians were admitted to the privileged classes in the city to fulfill numerous civic roles. Of course, it was essential in such cases for the Egyptians to become "Hellenized," to adopt Greek habits and the Greek language. Given that the Alexandrian mathematicians mentioned here were active several hundred years after the founding of the city, it would seem at least equally possible that they were ethnically Egyptian as that they remained ethnically Greek. In any case, it is unreasonable to portray them with purely European features when no physical descriptions exist.

  7. ^ D. M. Burton (1991, 1995). History of Mathematics, Dubuque, IA (Wm.C. Brown Publishers).

    Diophantos was most likely a Hellenized Babylonian.

  8. ^ Oswald Spengler (1923), Der Untergang des Abendlandes, 2 Bande:

    Were Plotin and Diophantus maybe of Jewish or Chaldaic origins?

  9. ^ J. Sesiano (1982). Books IV to VII of Diophantus' Arithmetica in the Arabic Translation Attributed to Qusta ibn Luqa. New York/Heidelberg/Berlin: Springer-Verlag. p. 502.
  10. ^ Hankel H., “Geschichte der mathematic im altertum und mittelalter, Leipzig, 1874. (translated to English by Ulrich Lirecht in Chinese Mathematics in the thirteenth century, Dover publications, New York, 1973.
  11. ^ a b Herrin, Judith (2013-03-18). Margins and Metropolis: Authority across the Byzantine Empire. Princeton University Press. p. 322. ISBN 140084522X.
  12. ^ "Diophantus biography". www-history.mcs.st-and.ac.uk. Retrieved 10 April 2018.
  13. ^ Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria, in: Historia Matematica, New York, 1993, Vol.20, No.2, 180-192
  14. ^ Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228
  15. ^ Kurt Vogel, "Diophantus of Alexandria." in Complete Dictionary of Scientific Biography, Encyclopedia.com, 2008.

References

  • Allard, A. "Les scolies aux arithmétiques de Diophante d'Alexandrie dans le Matritensis Bibl.Nat.4678 et les Vatican Gr.191 et 304" Byzantion 53. Brussels, 1983: 682-710.
  • Bachet de Méziriac, C.G. Diophanti Alexandrini Arithmeticorum libri sex et De numeris multangulis liber unus. Paris: Lutetiae, 1621.
  • Bashmakova, Izabella G. Diophantos. Arithmetica and the Book of Polygonal Numbers. Introduction and Commentary Translation by I.N. Veselovsky. Moscow: Nauka [in Russian].
  • 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.
  • Czwalina, Arthur. Arithmetik des Diophantos von Alexandria. Göttingen, 1952.
  • 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.
  • Rashed, Roshdi. L’Art de l’Algèbre de Diophante. éd. arabe. Le Caire : Bibliothèque Nationale, 1975.
  • Rashed, Roshdi. Diophante. Les Arithmétiques. Volume III: Book IV; Volume IV: Books V–VII, app., index. Collection des Universités de France. Paris (Société d’Édition “Les Belles Lettres”), 1984.
  • Sesiano, Jacques. The Arabic text of Books IV to VII of Diophantus’ translation and commentary. Thesis. Providence: Brown University, 1975.
  • 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, doi:10.1007/978-1-4613-8174-7.
  • Σταμάτης, Ευάγγελος Σ. Διοφάντου Αριθμητικά. Η άλγεβρα των αρχαίων Ελλήνων. Αρχαίον κείμενον – μετάφρασις – επεξηγήσεις. Αθήναι, Οργανισμός Εκδόσεως Διδακτικών Βιβλίων, 1963.
  • Tannery, P. L. Diophanti Alexandrini Opera omnia: cum Graecis commentariis, Lipsiae: In aedibus B.G. Teubneri, 1893-1895 (online: vol. 1, vol. 2)
  • Ver Eecke, P. Diophante d’Alexandrie: Les Six Livres Arithmétiques et le Livre des Nombres Polygones, Bruges: Desclée, De Brouwer, 1921.
  • Wertheim, G. Die Arithmetik und die Schrift über Polygonalzahlen des Diophantus von Alexandria. Übersetzt und mit Anmerkungen von G. Wertheim. Leipzig, 1890.

Further reading

  • Bashmakova, Izabella G. "Diophante et Fermat," Revue d'Histoire des Sciences 19 (1966), pp. 289-306
  • Bashmakova, Izabella G. Diophantus and Diophantine Equations. Moscow: Nauka 1972 [in Russian]. German translation: Diophant und diophantische Gleichungen. Birkhauser, Basel/ Stuttgart, 1974. English translation: Diophantus and Diophantine Equations. Translated by Abe Shenitzer with the editorial assistance of Hardy Grant and updated by Joseph Silverman. The Dolciani Mathematical Expositions, 20. Mathematical Association of America, Washington, DC. 1997.
  • Bashmakova, Izabella G. “Arithmetic of Algebraic Curves from Diophantus to Poincaré,” Historia Mathematica 8 (1981), 393-416.
  • Bashmakova, Izabella G., Slavutin, E.I. History of Diophantine Analysis from Diophantus to Fermat. Moscow: Nauka 1984 [in Russian].
  • Heath, Sir Thomas (1981). A history of Greek mathematics. 2. Cambridge University Press: Cambridge.
  • Rashed, Roshdi, Houzel, Christian. Les Arithmétiques de Diophante : Lecture historique et mathématique, Berlin, New York : Walter de Gruyter, 2013.
  • Rashed, Roshdi, Histoire de l’analyse diophantienne classique : D’Abū Kāmil à Fermat, Berlin, New York : Walter de Gruyter.
  • Vogel, Kurt (1970). "Diophantus of Alexandria". Dictionary of Scientific Biography. 4. New York: Scribner.

External links

Adequality

Adequality is a technique developed by Pierre de Fermat in his treatise Methodus ad disquirendam maximam et minimam (a Latin treatise circulated in France c. 1636) to calculate maxima and minima of functions, tangents to curves, area, center of mass, least action, and other problems in calculus. According to André Weil, Fermat "introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.11 shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings." (Weil 1973). Diophantus coined the word παρισότης (parisotēs) to refer to an approximate equality. Claude Gaspard Bachet de Méziriac translated Diophantus's Greek word into Latin as adaequalitas. Paul Tannery's French translation of Fermat’s Latin treatises on maxima and minima used the words adéquation and adégaler.

Algebra

Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in the letter is unknown, but applying additive inverses can reveal its value: . In E = mc2, the letters and are variables, and the letter is a constant, the speed of light in a vacuum. Algebra gives methods for writing formulas and solving equations that are much clearer and easier than the older method of writing everything out in words.

The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology.

A mathematician who does research in algebra is called an algebraist.

Arithmetica

Arithmetica (Greek: Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations.

Artsimovich (crater)

Artsimovich is a small lunar impact crater located in the western Mare Imbrium of the Earth's Moon. This is a circular crater forming a cup-shaped excavation in the surface of the lunar mare. It is named after Soviet physicist Lev A. Artsimovich. To the east is the crater Diophantus and to the northeast lies Delisle. Less than 20 kilometers to the north-northeast is the tiny Fedorov. This crater was identified as Diophantus A before being named by the International Astronomical Union.

Brahmagupta–Fibonacci identity

In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity says

For example,

The identity is also known as the Diophantus identity, as it was first proved by Diophantus of Alexandria. It is a special case of Euler's four-square identity, and also of Lagrange's identity.

Brahmagupta proved and used a more general identity (the Brahmagupta identity), equivalent to

This shows that, for any fixed A, the set of all numbers of the form x2 + A y2 is closed under multiplication.

The identity holds in the ring of integers, the ring of rational numbers and, more generally, any commutative ring. All four forms of the identity can be verified by expanding each side of the equation. Also, (2) can be obtained from (1), or (1) from (2), by changing b to −b, and likewise with (3) and (4).

Claude Gaspard Bachet de Méziriac

Claude Gaspard Bachet de Méziriac (9 October 1581 – 26 February 1638) was a French mathematician, linguist, poet and classics scholar born in Bourg-en-Bresse, at that time belonging to Duchy of Savoy.

Bachet was a pupil of the Jesuit mathematician Jacques de Billy at the Jesuit College in Rheims. They became close friends.

Bachet wrote the Problèmes plaisants, of which the first edition was issued in 1612, a second and enlarged edition was brought out in 1624; this contains an interesting collection of arithmetical tricks and questions, many of which are quoted in W. W. Rouse Ball's Mathematical Recreations and Essays. He also wrote Les éléments arithmétiques, which exists in manuscript; and a translation, from Greek to Latin, of the Arithmetica of Diophantus (1621). It was this very translation in which Fermat wrote his famous margin note claiming that he had a proof of Fermat's last theorem. The same text renders Diophantus' term παρισὀτης as adaequalitat, which became Fermat's technique of adequality, a pioneering method of infinitesimal calculus.

Bachet was the earliest writer who discussed the solution of indeterminate equations by means of continued fractions. He also did work in number theory and found a method of constructing magic squares. Some credible sources also name him the founder of the Bézout's identity.For a year in 1601 Bachet was a member of the Jesuit Order. He lived a comfortable life in Bourg-en-Bresse and married in 1612. He was elected member of the Académie française in 1635.

Delisle (crater)

Delisle is a small lunar impact crater in the western part of the Mare Imbrium. It was named after French astronomer Joseph-Nicolas Delisle. It lies to the north of the crater Diophantus, and just to the northwest of the ridge designated Mons Delisle. Between Delisle and Diophantus is a sinuous rille named Rima Diophantus, with a diameter of 150 km. To the northeast is another rille designated Rima Delisle, named after this crater.

The rim of Delisle is somewhat polygonal in form and it has a low central rise on the floor. There is some slight slumping along the inner wall, but overall the rim is still relatively fresh with little appearance of significant wear. The outer rim is surrounded by a small rampart of hummocky terrain.

This formation has also been designated "De l'Isle" in some sources.

Diophantine equation

In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is such that all the unknowns take integer values). A linear Diophantine equation equates the sum of two or more monomials, each of degree 1 in one of the variables, to a constant. An exponential Diophantine equation is one in which exponents on terms can be unknowns.

Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it.

The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.

While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations (beyond the theory of quadratic forms) was an achievement of the twentieth century.

Diophantus (crater)

Diophantus is a lunar impact crater that lies in the southwestern part of the Mare Imbrium. It was named after ancient Greek mathematician Diophantus. It forms a pair with the larger crater Delisle to the north. Diophantus has a wide inner wall and a low central rise. To the north of Diophantus is the sinuous rille designated Rima Diophantus, named after the crater. There is a tiny craterlet near the exterior of the southwest wall.

Diophantus (general)

Diophantus (Greek: Διόφαντος), son of Asclepiodorus, of Sinope, was a general in the service of Mithridates VI of Pontus. Diophantus was active in Mithridates' campaigns in the Bosporan Kingdom and elsewhere around the Black Sea, although their chronology is disputed. An inscription found during the excavations in Chersonesos glorifies Diophantus as "the first foreign invader to conquer the Scythians".During his first Crimean expedition, he relieved the siege of Chersonesos by the Scythian king Palacus and subdued his allies, the Tauri. He finished this campaign at Scythian Neapolis. During the second campaign, Diophantus checked another invasion of the Scythians, who had joined their forces with the Rhoxolanoi under Tasius. At one point during these campaigns he established a stronghold at Eupatorium on the eastern shore of the Crimea.

Around 107 BC, Mithridates dispatched Diophantes to Panticapaeum with the task of persuading the Bosporan king Paerisades V to cede his kingdom to Mithridates. While he was in the city, the Scythians, led by a certain Saumacus, revolted and killed Paerisades, while Diophantes barely managed to escape to Chersonesos. Back in Pontus, Diophantes rallied his forces and sailed to Crimea with a large fleet. The Scythian uprising was put down and the Bosporan kingdom was reduced to a dependency of Pontus.

Diophantus II.VIII

The eighth problem of the second book of Diophantus's Arithmetica is to divide a square into a sum of two squares.

Fedorov (crater)

Fedorov is a lunar geologic feature ("crater" in IAU nomenclature) located in the western Mare Imbrium. It was named after Russian rocket scientist A. P. Fyodorov. It lies east-northeast of the crater Diophantus, and southeast of Delisle. About 20 kilometers to the south-southeast is the slightly larger formation of Artsimovich.

This feature is slightly elongated and oddly shaped, with a ridge on the northern side. This ridge is about as large around the base as Federov crater, and rises about 0.8 km above the surrounding lunar mare.

History of algebra

Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra (in fact, every proof must use the completeness of the real numbers, which is not an algebraic property).

This article describes the history of the theory of equations, called here "algebra", from the origins to the emergence of algebra as a separate area of mathematics.

Metrodorus (grammarian)

Metrodorus (Greek: Μητρόδωρος; fl. c. 6th century) was a Greek grammarian and mathematician, who collected mathematical epigrams which appear in the Greek Anthology.

Nothing is known about the life of Metrodorus. The time he lived is not certain: he may have lived as early as the 3rd century AD, but it is more likely that he lived in the time of the emperors Anastasius I and Justin I, in the early 6th century.His name occurs in connection with 45 mathematical epigrams which are to be found in book 14 of the Greek Anthology. Although he may have authored some of the epigrams, it is generally accepted that he collected most of them, and some of them may predate the 5th century BC. Many of the epigrams lead to simple equations, and they are of the same type as those found in the Rhind Mathematical Papyrus (17th century BC). Among the problems Metrodorus collected are:

Twenty-three simple equations with one unknown, one of which is the famous epigram which reveals the age of Diophantus.

Twelve are easy simultaneous equations with two unknowns.

One gives a simultaneous equation with three unknowns.

Six are problems about filling and emptying vessels by pipes.

Number theory

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers).

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 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 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 preferred as an adjective to number-theoretic.

Paerisades V

Paerisades V (Greek: Παιρισάδης) was the son of Paerisades III and Kamasarye Philoteknos. He was last Spartocid ruler of the Bosporan Kingdom and ruled from 125 to 109 BC after the death of his brother Paerisades IV Philometor. With his death, ended a dynasty of Bosporan kings that had ruled the Bosporan Kingdom for over 3 centuries, starting in 438 BC with his ancestor Spartokos I.

Palacus

Palacus or Palakus was the king of Crimean Scythia who succeeded his father, Skilurus. Resuming the latter's war against Mithridates the Great, he attempted to besiege Chersonesos but was defeated by Pontic forces under Diophantus. Enlisting the assistance of the Rhoxolani under Tasius, Palacus launched an invasion of the Crimea. The invaders were defeated by Diophantus and accepted Mithridates as their overlord. Palacus was the last Scythian king whose name is attested in classical sources.

Papilio diophantus

Papilio diophantus is a species of swallowtail butterfly from the genus Papilio that is found in Sumatra.

Pierre de Fermat

Pierre de Fermat (French: [pjɛːʁ də fɛʁma]) (between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of 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 differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' Arithmetica.

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