''Diophantus and Diophantine Equations'' is a book in the

history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...

, on the history of

Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...

s and their solution by

Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...

of Alexandria. It was originally written in

Russian Russian(s) refers to anything related to Russia, including: *Russians (, ''russkiye''), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *Rossiyane (), Russian language term for all citizens and peo ...

Isabella Bashmakova Isabella Grigoryevna Bashmakova (russian: Изабелла Григорьевна Башмакова, 1921–2005) was a Russian historian of mathematics. In 2001, she was a recipient of the Alexander Koyré́ Medal of the International Academy ...

, and published by Nauka in 1972 under the title ''Диофант и диофантовы уравнения''. It was translated into German by Ludwig Boll as ''Diophant und diophantische Gleichungen'' (

Birkhäuser Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields: * Springer continues to publish science (particu ...

, 1974) and into English by Abe Shenitzer as ''Diophantus and Diophantine Equations'' (Dolciani Mathematical Expositions 20,

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

, 1997).

Topics

In the sense considered in the book, a

is an equation written using

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

s whose coefficients are

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

s. These equations are to be solved by finding rational-number values for the variables that, when plugged into the equation, make it become true. Although there is also a well-developed theory of

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

(rather than rational) solutions to polynomial equations, it is not included in this book.

of Alexandria studied equations of this type in the second century AD. Scholarly opinion has generally held that Diophantus only found solutions to specific equations, and had no methods for solving general families of equations. For instance,

Hermann Hankel Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician. Having worked on mathematical analysis during his career, he is best known for introducing the Hankel transform and the Hankel matrix. Biography Hankel was born on 1 ...

has written of the works of Diophantus that "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". In contrast, the thesis of Bashmakova's book is that Diophantus indeed had general methods, which can be inferred from the surviving record of his solutions to these problems. The opening chapter of the books tells what is known of Diophantus and his contemporaries, and surveys the problems published by Diophantus. The second chapter reviews the mathematics known to Diophantus, including his development of negative numbers, rational numbers, and powers of numbers, and his philosophy of mathematics treating numbers as

dimensionless quantities A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

, a necessary preliminary to the use of inhomogeneous polynomials. The third chapter brings in more modern concepts of

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

including the

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

and

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

of an

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...

, and

rational mapping In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. Definition Formal de ...

s and birational equivalences between curves. Chapters four and five concern

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...

s, and the theorem that when a conic has at least one rational point it has infinitely many. Chapter six covers the use of

secant line Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciproc ...

s to generate infinitely many points on a

cubic plane curve In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an eq ...

, considered in modern mathematics as an example of the

group law In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. The ...

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

s. Chapter seven concerns

Fermat's theorem on sums of two squares In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as: :p = x^2 + y^2, with ''x'' and ''y'' integers, if and only if :p \equiv 1 \pmod. The prime numbers for which this is true ar ...

, and the possibility that Diophantus may have known of some form of this theorem. The remaining four chapters trace the influence of Diophantus and his works through

Hypatia Hypatia, Koine pronunciation (born 350–370; died 415 AD) was a neoplatonist philosopher, astronomer, and mathematician, who lived in Alexandria, Egypt, then part of the Eastern Roman Empire. She was a prominent thinker in Alexandria wher ...

and into 19th-century Europe, particularly concentrating on the development of the theory of elliptic curves and their group law. The German edition adds supplementary material including a report by Joseph H. Silverman on progress towards a proof of

Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...

. An updated version of the same material was included in the English translation.

Audience and reception

Very little mathematical background is needed to read this book. Despite "qualms about Bashmakova's historical claims", reviewer David Graves writes that "a wealth of material, both mathematical and historical, is crammed into this remarkable little book", and he recommends it to any

number theorist Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

or scholar of the

. Reviewer Alan Osborne is also positive, writing that it is "well-crafted, ... offering considerable historical information while inviting the reader to explore a great deal of mathematics."

References

{{reflist, refs= {{citation, first=R., last=Bolling, title=Review of ''Диофант и диофантовы уравнения'', journal=

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

and

zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure mathematics, pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Informa ...

, language=German, mr=0414483, zbl=0241.01003 {{citation, title=Review of ''Diophantus and Diophantine Equations'', first=David, last=Graves, journal=MAA Reviews, publisher=

, date=February 1999, url=https://www.maa.org/press/maa-reviews/diophantus-and-diophantine-equations {{citation, title=Review of ''Diophantus and Diophantine Equations'', first=K.-B., last=Gundlach, journal=

, zbl=0883.11001, language=German {{citation, first=Hermann, last=Hankel, title=Zur Geschichte der Mathematik in Alterthum und Mittelalter, location=Leipzig, publisher=Teubner, year=1874, url=https://www.digitale-sammlungen.de/de/view/bsb11187597?page=174, pages=164–165, language=German. As translated in {{citation, title=Chinese Mathematics in the Thirteenth Century, first=Ulrich, last=Libbrecht, publisher=Dover, year=2005, isbn=9780486446196, page=218, url=https://books.google.com/books?id=hChTX2NlHUgC&pg=PA218 {{citation, last=Osborne, first=Alan, date=January 1999, issue=1, journal=The Mathematics Teacher, jstor=27970826, page=70, title=Review of ''Diophantus and Diophantine Equations'', volume=92 {{citation, first=R., last=Steiner, title=Review of ''Diophant und diophantische Gleichungen'', journal=

, mr=0485648 Diophantine equations Books about the history of mathematics 1972 non-fiction books 1974 non-fiction books 1997 non-fiction books