Lecture 13. Diophantus and Diophantine Equations

Lecture 13. Diophantus and Diophantine equations

Diophantus Diophantus of Alexandria, (about 200 - 284), was a Greek mathematician. He was sometimes called “the father of algebra,” a title some claim should be shared by a Persian mathematician al-Khw¯arizm¯i (about 780 - 850).1

Figure 13.1 Fall of Rome

Because of catastrophic events such as the fall of Rome and the rise of Islam, and the burning of the library on Alexandria in 640, almost all details of Diophantus’ life and mathematical works were burned. The only clue to Diophantus’ life is a conundrum in the Greek Anthology (around 600 ), which is an algebraic problem: 2

1See Lecture 17. 2Victor J. Katz, A History of Mathematics - an introduction, 3rd edition, Addison -Wesley, p.172; or 2nd edition, p.168.

81 This tomb holds Diophantus ... (and) tells scientiﬁcally the measure of his life. God granted him to be a boy for the sixth part of his life, and adding a twelfth part to this. He clothed his cheeks with down. He lit the light of wedlock after a seventh part, and ﬁve years after his marriage, he granted him a son. Alas! lateborn wretched child; after attaining the measure of half his father’s life, chill Fate look him. After consoling his grief by this science of numbers for four years, he ended his life.

If this information is correct, by solving this algebraic problem, Diophantus married at 33 and had a son who died at 43, four years before Diophantus himself died at 84. Diophantus’ Arithmetica Diophantus was the author of three books, one is called the Arithmetica that deals with solving algebraic equations, while the other two books are now lost. For the Arithmetica, Diophantus tells us in his introduction that it is divided into thirteen books. The Arithmetica contains 189 problems. Only six have survived in Greek. Four were preserved by the Arabs and translated into Latin in the sixteenth century. Diophantus’ books had enormous inﬂuence on the development of number theory.

Figure 13.2 Diophantus’ Arithmetica.

82 It is well-known that Pierre de Fermat studied Arithmetica and made a fateful note in the margin of his copy of the book that a certain equation similar to the Pythagorean equation

xn + yn = zn, n ≥ 3 considered by Diophantus has no solutions and he found “a truly marvelous proof of this proposition. This is the celebrated Fermat’s Last Theorem (see Lecture 23). This led to tremendous advances in number theory, and the study of Diophantine equations (“Diophan- tine geometry”) and of Diophantine approximations remain important areas of mathematical research.

Diophantus studied algebra, which was avoided by many Greek mathematicians at the time because of the discovery of irrational numbers and their inability to handle them.

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.

Figure 13.3 A map of Alexandria in 81 AD

Some of Diophantus’ mathematical work

• All of Diophantus’ symbols are abbreviations, which is one of his important achieve- ments. Before him, people used words. For example, he called “unknown” as the number of the problem, used ψ to denote “-”, and τ to denote “=”, and used algebraic symbols to denote “square” and “cube” of numbers.

83 • Diophantus did know how to solve quadratic equations of the form ax2 + c = bx. He only took positive roots. One example is: To ﬁnd two numbers such that their sum and the sum of their squares are given numbers. In modern notation, given a, b, it asked to ﬁnd x, y such that

x + y = a, x2 + y2 = b.

• He knew “negative times negative is positive.”

• Most of Diophantus’ problems are indeterminants, i.e., the number of equations is more than the number of unknowns. For these problems, Diophantus generally gives only one solution, but one may extend the method to obtain other solutions.

• He studied equations of higher degree. For example, in modern notation, to ﬁnd x, y and z such that (x2)2 +(y3)2 = z2.

Another example, to ﬁnd x and y such that

x + y = a, x3 + y3 = b.

• Diophantus knew the expansion of (x + y)3.

Let us take look at one problem quoted from Diophantus 3:

Let it be required to divide 16 into two squares. And let the ﬁrst square = x2; then the other will be 16 − x2; it shall be required therefore to make 16 − x2 =a square. I take a square of the form (ax − 4)2, a being any integer and 4 the root of 16; for example, let the side be 2x − 4, and the square itself 4x2 + 16 − 16x. Then 4x2 + 16 − 16x = 16 − x2. To add to both sides the negative terms and 2 16 take like from like, Then 5x = 16x, and x = 5 . One number will therefore be 256 144 400 25 , the other 25 , and their sum is 25 or 16, and each is a squre. In other words, it is in fact an indeterminant problem: To ﬁnd rational solutions of the equation x2 +y2 = 16. Set y = ax−4. Substituting into the equation, we get x2 +(ax−4)2 = 8a 16, i.e., x = 1+a2 .

3Victor J. Katz, A History of Mathematics - an introduction, 3rd edition, Addison -Wesley, 2009, p.179; or 2nd edition, p.177.