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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 x2 + y2 = z2. 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, 2n2 + 2n, 2n2 + 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, n2 - 1, n2 + 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

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.