# Number Theory: Fermat's Last Theorem

Exercise 4.7: A primitive Pythagorean triple is one in
which any two of

the three numbers are relatively prime. Show that every multiple of a
Pythagorean

triple is again a Pythagorean triple, and that every Pythagorean

triple is a multiple of a primitive one.

Exercise 4.8: Show that the sum of two odd squares is
never a square,

and use this fact to conclude that all Pythagorean triples have an even leg.

Exercise 4.9: Look up the Euclidean algorithm and use it
to decide

whether a Pythagorean triple is primitive or not.

Exercise 4.10: Show that FLT is true for all exponents n
if it is true for

n = 4 and all odd prime numbers n.

## 4.2 Euclid’s Classification of Pythagorean Triples

A triple of positive integers (x, y, z) is called a
Pythagorean triple if the

integers satisfy the equation x^{2} + y^{2} = z^{2}. Such a triple is called primitive

if x, y, z have no common factor. For instance, (3, 4, 5) and (5, 12, 13) are

primitive triples, whereas (6, 8, 10) is not primitive, but is a Pythagorean

triple. The significance of primitive triples is that “multiples” of primitive

ones account for all triples (see Exercise 4.7 in the previous section).

The problem of finding Pythagorean triples occupied the minds of mathematicians

as far back as the Babylonian civilization. Analysis of cuneiform

clay tablets shows that the Babylonians were in possession of a systematic

method for producing Pythagorean triples [127, pp. 36 ff.].

For instance, the tablet catalogued as Plimpton 322 in
Columbia University’s

Plimpton Collection, dating from 1900–1600 b.c.e., contains a list of

fifteen Pythagorean triples as large as (12709, 13500, 18541). (Is this triple

primitive?) For a detailed discussion of this tablet see, e.g., [64], [129].
There

is reason to believe that the Babylonians might even have known the complete

solution to the problem [131, pp. 175–79]. Other civilizations, such

as those of China and India, also have studied the problem [93]. Clearly,

Pythagorean triples are related to geometry via the Pythagorean Theorem,

as a Pythagorean triple corresponds to a right triangle with integer sides.

The Pythagoreans, after whom the theorem is named, were an
ancient

Greek school that flourished around the sixth century B.C.E. Aristotle says

that they “applied themselves to the study of mathematics, and were the

first to advance that science, insomuch that, having been brought up in it,

they thought that its principles must be the principles of all existing things”

[85, p. 36]. Their motto is said to have been “all is number” [20, p. 54].

The particular interest of the Pythagoreans in relationships between whole

numbers naturally led to the investigation of right triangles with integral

sides. Proclus, a later commentator, who taught during the fifth century

c.e. at the Neo-Platonic Academy in Athens, credits the Pythagoreans

**PHOTO 4.7. Plimpton 322.
**with the formula

(2n + 1, 2n

^{2}+ 2n, 2n

^{2}+ 2n + 1) ,

for any positive integer n, generating triples in which
the longer leg and

hypotenuse differ by one. He also describes another formula,

(2n, n^{2} - 1, n^{2} + 1)

for all n ≥ 2, attributed to Plato [58, pp. 212–13], [85,
p. 47], [134, p. 340]

(Exercise 4.18).

The first formula giving a complete classification of all
triples appears in

Euclid’s Elements:

To find two square numbers such that their sum is also
square [51,

vol. 3, p. 63].

This is the first text that we will examine.

Very little is known about Euclid’s life. The most that
can be said with

any certainty is that he lived about 300 B.C.E. and taught at the University

in Alexandria. He probably attended Plato’s Academy in Athens,

which was at the time the center of Greek mathematics, and he was invited

to Alexandria when the great library and museum were being set up at the

direction of Ptolemy Soter. Under the rule of Ptolemy, Alexandria became

a world center both in commerce and scholarship, Euclid was a founder of

the great school of mathematics there. Proclus tells us that Euclid “put

together the Elements, collecting many of Eudoxus’s theorems, perfecting

many of Theaetetus’s, and also bringing to irrefragable demonstration the

things which were only somewhat loosely proved by his predecessors” [85,

p. 202]. There is an extensive entry on Euclid in [42], which is highly
recommended.

The Elements was intended as an introductory textbook on

elementary mathematics for students at the University. It ended up as the

**PHOTO 4.8. Pythagoras.
**all-time number one mathematical bestseller, and was used as a textbook

in a number of countries even into this century. We quote from the preface

of one such textbook [170] used in England during the second half of the

nineteenth century:

In England, the text-book of Geometry consists of the
Elements

of Euclid, for nearly every official programme of instruction or examination

explicitly includes some portion of this work. Numerous

attempts have been made to find an appropriate substitute for the

Elements of Euclid, but such attempts, fortunately, have hitherto

been made in vain. The advantages attending a common standard of

reference in such an important subject, can hardly be overestimated,

and it is extremely improbable, if Euclid were once abandoned, that

any agreement would exist as to the author who should replace him.

The Elements consists of thirteen books, which deal with a
variety of

different subjects. No manuscripts from Euclid’s time are known, and the

versions of the Elements in use today are translations of Arabic, Greek, and

Latin versions of various time periods. The first printed edition of the
Elements

was published at Venice in 1482. It was the first printed mathematics

book of any importance. For more details on the various translations see

**PHOTO 4.9. First printed edition of Euclid, 1482.
**[85, Ch. IX]. For a more detailed discussion of the structure of the
Elements

see the geometry chapter.

Euclid’s solution to the problem of the classification of
all Pythagorean

triples appears as Lemma 1 in Book X, before Proposition 29. Book X

concerns itself primarily with the theory and application of irrationals, and

is the first major treatise we have on this subject. Pythagorean triples

are used in Proposition 29 to produce examples of irrationals with certain

properties.

The reader should be cautioned that the following text is
rather subtle

and requires very careful reading. It is followed by an extensive mathemati-

cal commentary. In order to understand it, one needs to know what similar

plane numbers are, the central concept in Euclid’s argument. First of all,

a plane number is simply a formal product of two numbers, thought of as

the sides of a rectangle, of which the plane number itself gives the area.

Two plane numbers are similar if their respective sides are proportional

with the same rational proportionality factor. Geometrically, the two plane

numbers represent similar rectangles. For example, the two plane numbers

6 ٠ 10 and 9 ٠ 15 are similar.