Diophantus of Alexandria

Διόφαντος ο Αλεξανδρεύς

Diophant von Alexandrien

We know almost nothing about Diophantus, except that he lived, in Alexandria. We know that he quotes Hypsicles (150 BC) and he is quoted by Theon of Alexandria (whose date is fixed by the solar eclipse of June 16, 364 AD ).

This tomb hold Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life. J R Newman (ed.) The World of Mathematics (New York 1956).

From his epitaph, which can be written as a linear equation

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

(where x is the number of years of Diophantus lived ) we find a solution x = 84. We know therefore that he married at the age of 33, had a son who died when he was 42, 4 years before Diophantus himself died. Assumimg that he was born around 200 AD then Diophantus lived in the period (200 - 284 ) AD .

Diophantus began the systematic study of equations with integer coefficients. He was a pioneer in solving certain indeterminate algebraic equations, i.e., an equation in which the variables can take on integer values (Diophantine equations) and has an infinite but denumerable set of solutions: e.g., x+2y=3.

He published his results in the book Arithmetica. Other work: On polygonal numbers only fragments survived, the book Porismata was lost.

For the Diophantine equations of the form ax + by = c Euclid's algorithm provides an answer if a solution exists (integer x, y). If d is the greatest common divisor of a,b and c can be divided by d then such a solution exist.

Take the equation 6x+15y = 12. Then the greatest common divisor of 6 and 15 is 3 and 12 can by divided by 3. A solution exists and if we try we will find (x = -3 and y =2 which gives -18 + 30 = 12).

Did Diophantus know negative numbers? He says in Greek: Leiyis epileiyin pollaplasiasqeisa poiei uparcin, leiyis de epi uparcin poiei leiyin. (Book I, 20). Translated this means: minus times minus is plus and minus times plus is minus.

The second chapter comments that Diophantus extended the notion of number to include negatives and rationals, describes his symbols for exponents from -6 to 6, and notes that he moved beyond Greek traditions in permitting addition of non-homogeneous magnitudes. (Review by D. Graves of a book of I. G. Bashmakova about the Arithmetica)

Diophantus Symbols

ς

x

ΔY

x2 (dynamis) ( δύναμις)

KY

x3 (kybos) ( κύβος)

ΔYΔ

x4 (dynamodynamis) ( δυναμοδύναμις)

ΔKY

x5 (dynamokybos) ( δυναμόκυβος)

KYK

x6 (kybokybos) (κυβόκυβος )

Diophantus used for negative exponents the symbol X. For example ςX for 1/x = x -1.

was used for the minus sign (a symbol derived from Lambda, I and leiyis), was used for the constant term.

KY αςβγ for example means x3 +2x -3.

Almost 1500 years later François Viète (1540-1603) improved the notation of equations. See also The Father of Modern Algebraic Notation

In number theory Diophantus discovered that numbers of the form 4n + 3 cannot be the sum of two squares and numbers of the form 24n + 7 cannot be the sum of three squares. It is suggested that he know that every number can be written as the sum of 4 squares, but it is unlikely to have known a proof as Langrange provided much later. From Fragments of his book On polygonal numbers we know a few lemmata such as that given any numbers a, b then there exist numbers c, d such that a3 - b3 = c3 + d3.

Diophantus found three rationals 3/10 , 21/5 , 7/10 with the property that the product of any two of them increased by the sum of those two gives a perfect square [1] and Euler found four rationals 65/224 , 9/224 , 9/56 , 5/2 with the same property.

Diophantus provided also an algebraic equation of third degree for the problem

(x-1)3 = (x+1)2 + 2 for which he gave without any explanation a solution x = 4.

More than a thousand years later, in 1570, 6 of the 13 chapters of Arithmetica were found in a library in Germany. The work was translated first by Bombelli but not published. Claude Bachet’s translation into Latin in 1621 came into the hands of Fermat in Toulouse. Diophantus presents a collection of 189 problems, and their solution. Typically, one is asked to find rational numbers (fractions) satisfying some equation in two unknowns. A random example, Problem 24 of Book VI, asks to split the number 6 into two parts, say y and 6−y so that the product of those parts is the difference of a cube minus its cube root x. [The smallest nontrivial solution is x = 17/9, y = 26/27].

Fermat studies Arithmetica and among other discoveries he claims that he has a beautiful solution for the well known Last Theorem but he has not enough place to provide the solution as his son later tells us.

In a Lecture delivered before the International Congress of Mathematicians at Paris in 1900 David Hilbert, one of the greatest Mathematicians of the last century, provided a set of important mathematical problems.

Among these is the 10-th Problem: Determination of the Solvability of a Diophantine Equation.

Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: ”To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.

So Hilbert was asking for an algorithm or a program that can solve any possible Diophantine Equation.

For around 70 years Mathematicians tried to solve this problem. It is a journey from Turing machines to Fibonacci numbers that are related to the Golden section. Then in 1970 Yuri Matiyasevich solved the 10-th Problem of Hilbert showing that such an algorithm that can solve any Diophantine equation does not exist!

A special Diophantine equation is Pell’s equation x2 – Dy2 = 1

The Importance was that its solution is the main step in solution of general quadratic Diophantine equations in two variables and useful for Matiyasevich theorem on non-existence of the general algorithm for solving Diophantine equations.

The simplest case x2 – 2y2 = 1 was studied by Pythagoreans: if x and y are large solutions then x/y ≈ √2. We know that such a solution does not exist. Another special problem was the solution of x2 – 4729494y2 = 1 in Archimedes Cattle Problem for which we know a solution with a number with 206545 digits since 1880 (another restriction of the Cattle problem is that y must be a multiple of 9304).

References

History of Mathematics Lecture by Alexandre Karassev

  1. Diophantus of Alexandria. Arithmetics and the Book of Polygonal Numbers, (I. G. Bashmakova, Ed.), Moscow: Nauka, 1974 (in Russian).
  2. A. Dujella. ”On Diophantine quintuples.” Acta Arith. 81 (1997):69–79.
  3. A. Dujella. ”An extension of an old problem of Diophantus and Euler.” Fibonacci Quart. 37 (1999):312–314.
  4. J. Tropfke, Geschichte der Elementarmathematik, Volume 3, Walter de Gruyter, 1980

    DIOPHANTUS

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