Diophantus

Diophantus of Alexandria was a Greek mathematician who was the author of the Arithmetica in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations.
Joseph-Louis Lagrange called Diophantus "the inventor of algebra"; his exposition became the standard within the Neoplatonic schools of Late antiquity, and its translation into Arabic in the 9th century AD and had influence in the development of later algebra: Diophantus' method of solution matches medieval Arabic algebra in its concepts and overall procedure. The 1621 edition of Arithmetica by Bachet gained fame after Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy.
In modern use, Diophantine equations are algebraic equations with integer coefficients for which integer solutions are sought. Diophantine geometry and Diophantine approximations are two other subareas of number theory that are named after him. Some problems from the Arithmetica have inspired modern work in both abstract algebra and number theory.

Biography

The exact details of Diophantus' life are obscure. Although he probably flourished in the third century CE, he may have lived anywhere between 170 BCE, roughly contemporaneous with Hypsicles, the latest author he quotes from, and 350 CE, when Theon of Alexandria quotes from him. Paul Tannery suggested that a reference to an "Anatolius" as a student of Diophantus in the works of Michael Psellos may refer to the early Christian bishop Anatolius of Alexandria, who may possibly the same Anatolius mentioned by Eunapius as a teacher of the pagan Neopythagorean philosopher Iamblichus, either of which would place him in the 3rd century CE.
The only definitive piece of information about his life is derived from a set of mathematical puzzles attributed to the 5th or 6th century CE grammarian Metrodorus preserved in book 14 of the Greek Anthology. One of the problems 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 can be expressed as
which gives a value of 84 years. However, the accuracy of the information cannot be confirmed.

''Arithmetica''

Arithmetica is the major work of Diophantus and the most prominent work on premodern algebra in Greek mathematics. It is a collection of 290 algebraic problems giving numerical solutions of determinate equations and indeterminate equations. Arithmetica was originally written in thirteen books, but only six of them survive in Greek, while another four books survive in Arabic, which were discovered in 1968. The books in Arabic correspond to books 4 to 7 of the original treatise, while the Greek books correspond to books 1 to 3 and 8 to 10.
Arithmetica is the earliest extant work present that solve arithmetic problems by algebra. Diophantus however did not invent the method of algebra, which existed before him. Algebra was practiced and diffused orally by practitioners, with Diophantus picking up technique to solve problems in arithmetic.
Equations in the book are presently called Diophantine equations. The method for solving these equations is known as Diophantine analysis. Most of the Arithmetica problems lead to quadratic equations.

Notation

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.
Similar to medieval Arabic algebra, Diophantus uses three stages to solution of a problem by algebra:

An unknown is named and an equation is set up
An equation is simplified to a standard form
Simplified equation is solved

Diophantus does not give classification of equations in six types like Al-Khwarizmi in extant parts of Arithmetica. He does say that he would give solution to three terms equations later, so this part of work is possibly just lost.
The main difference between Diophantine notation and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials. So for example, what would be written in modern notation as
which can be rewritten as
would be written in Diophantus's notation as

Symbol	What it represents
	1
	2
	5
	10
ἴσ	"equals"
	represents the subtraction of everything that follows up to ἴσ
	the zeroth power
	the unknown quantity
	the second power, from Greek δύναμις, meaning strength or power
	the third power, from Greek κύβος, meaning a cube
	the fourth power
	the fifth power
	the sixth power

Unlike in modern notation, the coefficients come after the variables and addition is represented by the juxtaposition of terms. A literal symbol-for-symbol translation of Diophantus's equation into a modern equation would be the following:
where to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:

In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers. He also noticed that numbers of the form cannot be the sum of two squares. Diophantus also appears to know that every number can be written as the sum of four squares. If he did know this result, his doing so would be truly remarkable: even Fermat, who stated the result, failed to provide a proof of it and it was not settled until Joseph-Louis Lagrange proved it using results due to Leonhard Euler.

Other works

Another work by Diophantus, On Polygonal Numbers is transmitted in an incomplete form in four Byzantine manuscripts along with the Arithmetica. Two other lost works by Diophantus are known: Porisms and On Parts.
Recently, Wilbur Knorr has suggested that another book, Preliminaries to the Geometric Elements, traditionally attributed to Hero of Alexandria, may actually be by Diophantus.

On polygonal numbers

This work on polygonal numbers, a topic that was of great interest to the Pythagoreans consists of a preface and five propositions in its extant form. The treatise breaks off in the middle of a proposition about how many ways a number can be a polygonal number.

The ''Porisms''

The Porisms was a collection of lemmas along with accompanying proofs. Although The Porisms is lost, we know three lemmas contained there, since Diophantus quotes them in the Arithmetica and refers the reader to the Porisms for the proof.
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 and, with, there exist, all positive and rational, such that

''On Parts''

This work, on fractions, is known by a single reference, a Neoplatonic scholium to Iamblichus' treatise on Nicomachus' Introduction to Arithmetic. Next to a line where Iamblichus writes "Some of the Pythagoreans said that the unit is the borderline between number and parts" the scholiast writes "So Diophantus writes in On Parts, for parts involve progress in diminution carried to infinity."

Influence

Diophantus' work has had a large influence in history. Although Joseph-Louis Lagrange called Diophantus "the inventor of algebra", he did not invent it, however his work Arithmetica created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra. Diophantus and his works influenced mathematics in the medieval Islamic world, and 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.

Later antiquity

After its publication, Diophantus' work continued to be read in the Greek-speaking Mediterranean from the 4th through the 7th centuries. The earliest known reference to Diophantus, in the 4th century, is the Commentary on the Almagest Theon of Alexandria, which quotes from the introduction to the Arithmetica. According to the Suda, Hypatia, who was Theon's daughter and frequent collaborator, wrote a now lost commentary on Diophantus' Arithmetica, which suggests that this work may have been closely studied by Neoplatonic mathematicians in Alexandria during Late antiquity. References to Diophantus also survive in a number of Neoplatonic scholia to the works of Iamblichus. A 6th century Neoplatonic commentary on Porphyry's Isagoge by Pseudo-Elias also mentions Diophantus; after outlining the quadrivium of arithmetic, geometry, music, and astronomy and four other disciplines adjacent to them, it mentions that Nicomachus occupies the first place in arithmetic but Diophantus occupies the first place in "logistic", showing that, despite the title of Arithmetica, the more algebraic work of Diophantus was already seen as distinct from arithmetic prior to the medieval era.

Medieval era

Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the 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 are preserved together with a comprehensive commentary written by the earlier Greek scholar Maximos Planudes, who produced an edition of Diophantus within the library of the Chora Monastery in Byzantine Constantinople.
Arithmetica became known to mathematicians in the Islamic world in the ninth century, when Qusta ibn Luqa translated it into Arabic.
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.