# Number Theory: Fermat's Last Theorem

## Euclid, from

Elements

**LEMMA 1
(before Proposition 29 in Book X)
**To find two square numbers such that their sum is also a square.

Let two numbers AB , BC be set out, and let them be either both even or

both odd.

Then since, whether an even number is subtracted from an
even number,

or an odd number from an odd number, the remainder is even [IX. 24, 26],

therefore the remainder AC is even.

Let AC be bisected at D.

Let AB, BC also be either similar plane numbers, or square
numbers , which

are themselves also similar plane numbers.

Now the product of AB , BC together with the square on CD
is equal to

the square on BD [II. 6].

And the product of AB, BC is square, inasmuch as it was
proved that, if

two similar plane numbers by multiplying one another make some number the

product is square [IX. 1].

Therefore two square numbers, the product of AB, BC, and
the square on

CD, have been found which, when added together , make the square on BD.

And it is manifest that two square numbers, the square on
BD and the

square on CD, have again been found such that their difference, the product

of AB, BC, is a square, whenever AB, BC are similar plane numbers.

But when they are not similar plane numbers, two square
numbers, the

square on BD and the square on DC, have been found such that their

difference, the product of AB, BC, is not square.

Q. E. D.

In order to translate Euclid’s argument into modern
algebraic notation,

let AB = u, BC = v, CD = d, BD = f. From [II.6] Euclid obtains the

equation

uv + d^{2} = f^{2},

which alternatively follows via

uv = (f + d)(f - d) = f^{2} - d^{2} ,

since D bisects AC and thus AD = CD = d.

The remainder of Euclid’s argument involves similar plane
numbers. One

can prove that two numbers are similar plane numbers if and only if their

ratio is the square of a rational number (see the lemma at the end of the

section). Euclid continues his analysis by noting that uv is a square if and

only if u and v are similar plane numbers, which follows from the previous

sentence. So far he has given a procedure that beginning with two similar

plane numbers of the same parity constructs a Pythagorean triple. (The

parity of a number is simply whether it is even or odd. Thus 4 is of even

parity. That two numbers are of the same parity means that they are either

both even or both odd.) (Observe that his plane numbers must be unequal,

why?)

But in fact all Pythagorean triples arise in this way, as
he somewhat

cryptically explains in the last paragraph. Start with a difference f ^{2} - d^{2},

and factor it as uv, where u = f + d and v = f - d. Since this u and v are

clearly of the same parity, Euclid’s construction can proceed using these

numbers. Notice that this will reproduce the original d and f with which

we started. The result is the equation uv = f^{2} -d^{2} as above, but as Euclid

says in his last paragraph, this difference of squares, which is uv, will only

itself be a square if u and v were similar plane numbers to begin with.

Thus Euclid has set up a one-to-one correspondence between differences of

squares and pairs of unequal numbers of the same parity, in such a way that

differences of squares that are themselves square correspond to pairs that

are similar plane numbers. Thus he has classified all Pythagorean triples in

terms of similar plane numbers, so the problem of finding all Pythagorean

triples reduces to that of finding all pairs of similar plane numbers of the

same parity (Exercises 4.12–4.13).

Now we can see how to find the formulas of the
Pythagoreans and Plato

that we mentioned. Since

we have

The simplest way to choose u, v such that their product is
a square is to

let u = k^{2} and v = 1. For f and d to be integers, we must have k odd, thus

leading to the formula of the Pythagoreans. Letting u = 2k^{2} and v = 2

gives the Platonic formula [51, Vol. 1, p. 385].

From Euclid’s classification of Pythagorean triples we can
derive the following

classification of all primitive Pythagorean triples, which we will need

for our next source. Suppose that the triple (d, e, f) is primitive. One of the

numbers d, e has to be even (this follows from Exercise 4.8 in the
Introduction),

let us assume that it is e, by interchanging d and e if necessary. Now,

as noted above, we can apply Euclid’s method to this triple, obtaining

f^{2} - d^{2} = e^{2} = uv,

with u, v similar plane numbers of the same parity. Thus u
and v are both

even, and it follows from the lemma below that they are of the form u =

mp^{2}, v = mq^{2}. A common divisor of u and v also divides u + v = 2f and

u - v = 2d. But since d and f are relatively prime, that is, their greatest

common divisor is 1, it follows that the greatest common divisor of u and

v is 1 or 2. Together with the fact that u and v are even, this implies that

m = 2, and that p and q are relatively prime. Thus, the triple is of the form

d = p^{2} - q^{2}, e = 2pq, f = p^{2} + q^{2} ,

and moreover, p and q have different parity , since d is
odd by primitivity

of the triple.

To summarize, any primitive triple has this form (by
interchanging d and

e if necessary), where p is greater than q, and p and q are relatively prime

and of different parity. Furthermore, any such choice of p and q results in

a primitive triple (check this).

**Lemma:** Two plane numbers are similar if and only if
their ratio is the

square of a rational number.

**Proof. **Suppose that u and v are similar plane
numbers. Then u = xy and

v = zw for some numbers (that is, positive integers ) x, y, z, w, which are

the sides of u and v, respectively. For these sides to be proportional means

that x = ry and z = rw for some rational number r. So

and y/w is clearly rational. Conversely, suppose u/v =
(p/q)^{2}, where we

may assume that p and q are relatively prime. So

As u is an integer, we know that q^{2} divides p^{2}v. Since p
and q are relatively

prime, q^{2} must divide v, that is, v = mq^{2} for some positive integer m.

Similarly, p^{2} must divide u, and it is easy to see that in fact u = mp^{2}. If

we choose p and mp to be the sides of u, and q and mq to be the sides of

v, we see that u and v are similar plane numbers.

Exercise 4.11: Prove that there are only a finite number
of Pythagorean

triples with a fixed leg.

Exercise 4.12: Show that there are exactly two pairs of
similar plane

numbers of same parity that will produce a given Pythagorean triple using

Euclid’s construction.

Exercise 4.13: Enumerate (that is, give a recipe for
listing) all pairs of

similar plane numbers with same parity. (Remember to show that your list

is complete.) Use your list to enumerate all Pythagorean triples.

Exercise 4.14: Enumerate all primitive Pythagorean triples.

Exercise 4.15: Derive a general formula from Diophantus’s
treatment of

Pythagorean triples [86], and compare it with Euclid’s.

Exercise 4.16: Give a geometric demonstration of Euclid’s
statement that

“the product of AB, BC together with the square on CD is equal to the

square on BD”, that is,

AB ٠ BC + (CD)^{2} = (BD)^{2}:

[Hint: Recall that the product of two numbers may be
represented geometrically

as the area of a rectangle with those numbers as the lengths of its

sides.]

Exercise 4.17: Enumerate as many Pythagorean triples as
you can that

contain the number 1378.

Exercise 4.18: In the two formulas for Pythagorean
triples, ascribed to

the Pythagoreans and Plato, respectively, which triples, if any, appear on

both lists? Are the triples generated by these lists primitive? Which are,

and which are not? Are there an infinite number of primitive triples? Do

the primitive triples in the above two lists exhaust all possible primitive

triples?

## 4.3 Euler’s Solution for Exponent Four

Without doubt Leonhard Euler is one of the world’s
mathematical giants ,

whose work profoundly transformed mathematics. He made extensive contributions

to many mathematical subjects, including number theory, and

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