Fermat's One Proof

In his life, Pierre de Fermat only left one proof in relation to number theory. He used his method of infinite descent to show that the area of a right triangle cannot be a square within the domain of whole numbers.

The proof itself was found after his death in his notes on Diophantus's Arithmetica.

In what follows, I will go through each sentence of Fermat's proof and provide details to make his step clear. My goal is to shed light on Fermat's thinking in his one existent proof.

I will show how this proof can also be used to prove the case of Fermat's Last Theorem where n = 4. In my next blog, I go over a simpler proof for n=4.

My analysis is based on the work by Harold M. Edwards called Fermat's Last Theorem. The translation is taken from T. L. Heath's translation.

Fermat writes:

(1) Fermat: "If the area of a right-angled triangle were a square, there would exist two biquadrates the difference of which would be a square number."

A biquadrate is a value to the fourth-power. So, the biquadrate of 2 is 2⁴ = 16.

This corresponds to the following equation:
p⁴ - q⁴ = z².

When I first read this, it was surprising to me that Fermat assumed that this equation was obvious from the problem of showing that right triangle's area cannot be equal to a square.

This equation proves n = 4 since if x⁴ + y⁴ = z⁴, then:
x⁴ = (x²)² = z⁴ - y⁴

So, proving there is no right triangle that has an area equal to a square will also prove Fermat's Last Theorem for n=4.

The steps to this equation can be traced as follows:

(a) A right triangle is characterized by the Pythagorean Theorem:
a² + b² = c².

(b) The area of a rectangle is base x height.

(c) A right triangle of a given base and height is created by dividing the same rectangle across the diagonal.

(d) Therefore, the area of a right triangle is base * height / 2.

(e) Or, using a,b,c from above: area = ab/2

(f) From the solution to Pythagorean Triples, we know that:
a = (2pq)d
b = (p² - q²)d
where we know that p,q are relatively prime.

(g)So, saying the area of a right triangle is equal to a square comes down to proving that there are no solutions for:
z² = ((2pq)d[p² - q²]d)/2 = (pq)(d²)[p² - q²]

(h) From a previous result, we know that d will divide z so, we are left with showing no solutions for:
(z/d)² = (pq)[p² - q²]

(i) Since p,q are relatively prime, it follows that (pq) and [p² - q²] are relatively prime [see here for details]

(j) From this we conclude, that pq is a square and p² - q² are squares.

(k) And since p,q are relatively prime, p,q are themselves squares.

(l) So, there exist P,Q such that p = P² and q = Q²

(m) Since pq is a square, there exists a value k such that k² = pq.

(n) And from (m), k divides (z/d), so we get:

(z/dk)² = p² - q² = P⁴ - Q⁴

(2) Fermat: "Consequently there would exist two square numbers the sum and difference of which would both be squares."

This one is a bit easier to derive
z² = P⁴ - Q⁴ = (P² + Q²)(P² - Q²)

Now, all, we need to show is that (P² + Q²) is relatively prime to (P² - Q²) which means that p + q is realtively prime to p - q. [ The trick is to remember that p,q are relatively prime and they are of different parity (see 13(b) here). See here for details]

In other words, there exist two square numbers (P²,Q²) the sum (P² + Q²) and difference (P² - Q²) are both squares.

(3) Fermat: "Therefore we should have a square number which would be equal to the sum of a square and the double of another square..."

We know that P² + Q² is equal to a square. Let's say S².
We also know that P² - Q² is equal to a square. Let's say T²
So that P² = Q² + T²
Which combined with the other equation gives us:
Q² + T² + Q² = T² + 2Q² = S²

We can assume that S,P, Q are relatively prime and also that P,T,Q are relatively prime with S,P,T odd and Q even. [See here for details]

We can also assume that Q,S,T are relatively prime. [See here for details]

(4) Fermat: "...while the squares of which this sum is made up would themselves have a square number for their sum."

And as stated before: P² = Q² + T².

(5) Fermat: "But if a square is made up of a square and the double of another square, its side, as I can very easily prove, is also similarly made up of a square and the double of another square."

So, he is saying this:
T² + 2Q² = S² --> S = t² + 2q².

This means that

2Q² = S² - T² = (S - T)(S + T)

And:

Q² = (1/2)(S-T)(S+T)

Now Q,S,T are relatively prime. We know that Q, S-T, S+T are even. [see above for details]
Let S-T = 2u, S + T = 2v
So that Q² = (1/2)(2u)(2v) = 2uv
Now, u,v are relatively prime [see here for details]
And, either u or v is even, let's assume u
So that, u = 2w and Q² = 2(2w)v
And w,v are relatively prime so w,v are squares.
Let w = W², v = V²
And S + T + S - T = 2S = 2u + 2v = 2(2W² + V²)
So that S = 2W² + V²

(6) Fermat: "From this we conclude that the said side is the sum of the sides about the right angle in a right-angled triangle and that the simple square contained in the sum is the base and the double of the other square is the perpendicular."

So, he is saying this:
S = 2W² + V² -> (2W² )² + ( V²)² is a square.

(2W² )² + ( V²)² = (2w)² + ( v)² = (u)² + ( v)² =
[(1/2)(S-T)]² + [(1/2)(S+T)]² =
[S² - 2ST + T²]/4 + [S² + 2ST + T²]/4 =
= [2S² + 2T²]/4 = (S² + T²)/2.

Now, since S² = 2Q² + T²
We have:
(2Q² + T² + T²)/2 = Q² + T²

Which equals P².

(7) Fermat: "This right-angled triangle will thus be formed from two squares, the sum and differences of which will be squares."

We can once again apply the Solution to the Pythagorean Triples so that:

2W²= 2mn
V² = m² - n²
P= m² + n²

So we've proven that the difference is a square. Now, we need to prove that the sum is a square.

Since Q² = 2(2w)v = 4W²V².

Q²= 2(2mn)(m² - n²)=4mn(m² - n²).

And

(Q/2)² = mn(m² - n²)

Since m,n and m² - n² are relatively prime, then m,n,m²-n² are all squares.

Letting m = M² and n = N²

We get:
V² = M⁴ - N⁴ = (M² - N²)(M² + N²)

Since M² - N², M² + N² are relatively prime, both the sum and differences are squares.

(8) Fermat: "But both these squares can be shown to be smaller than the squares originally assumed to be such that both their sum and differences are squares."

So M² + N² ≤ P² + Q².

Since M² + N² ≤ m + n is less than P which is less than P² + Q².

(9) Fermat: "Thus if there exist two squares such that their sum and differences are both squares, there will also exist two other integer squares which have the same property but a smaller sum"

We have proven that:
if P² + Q² is a square and P² - Q² is a square, then there exists M,N such that:
(a) M²+N² is less than P² + Q²
(b) M² + N² is a square
(c) M² - N² is a square.

(10) Fermat: "By the same reasoning we find a sum still smaller than the last found, and we can go on ad infinitum finding integer square numbers smaller and smaller which have the same property."

This leads to an infinite number of smaller solutions.

(11) Fermat: "This is, however, impossible because there cannot be an infinite series of numbers smaller than any given integer we please."

So that we have a proof by Infinite Descent.

(12) Fermat: "The margin is too small to enable me to give the proof completely and with all detail."

At least, this time we were able to reconstruct the proof. :-)

-Larry

Infinite Descent

Pierre de Fermat was very proud of his technique known as Infinite Descent. He wrote that this method "will lead to marvelous advancement in the theory of numbers" (quoted from: http://fermat.workjoke.com/flt2.htm)

As he often did, Fermat left very few details on how to apply the method. He did provide one example of this method in his proof that the area of a right triangle cannot be equal to a square number. An elegant application of this proof is found in the case of FLT: n=4 where the proof rests on the method of infinite descent and the solution to Pythagorean Triples.

The basic method is very straight forward. One proves that if an assumption is true, it means that the assumption must also be true for an element that is smaller. In other words, an assumption leads to the proposition of an infinite number of cases where it is true.

This technique is especially useful in the domain of positive integers. In this case, infinite descent is impossible since it contradicts the Well Ordering Principle. In other words, there must always be a smallest element in any set of positive integers. But if there is a smallest element, then, there cannot be an infinite descent. In this way, the method can be used as way to prove by contradiction certain negative assumptions. It can also be used to prove a positive conclusion as I will show below.

Fermat is known to have used this technique to prove that there is no square equal to a right triangle. He left a proof using this technique for the case of n = 4 which I will go over in a future blog. He also wrote about its use in the proof for the case of n = 3 in a letter to Christian Huygens.

If Fermat had really found a proof for his theorem, it was without doubt based on this method.

To show the technique in action, I will use it to prove the following theorem:

Thereom: Relatively Prime Divisors of an n-power are themselves n-powers.

This theorem says that if gcd(v,w) = 1 and vw = zⁿ
Then, there exists x,y such that v = xⁿ, w = yⁿ

(1) So, we start with gcd(v,w) = 1, vw = zⁿ
(2) Assume that v is not equal to any number xⁿ
(3) v ≠ 1 since 1 is an xⁿ power
(4) Now, v is divisible by a prime number p. [Fundamental Theorem of Arithmetic]
(5) So, there exists k such that v = pk
(6) p divides z since zⁿ = vw = pkw [By applying Euclid's Lemma]
(7) So, there exists m such that z=pm
(8) So, zⁿ = vw = pkw = (pm)ⁿ = pⁿmⁿ
(9) Dividing p from both sides gives us:
kw = p^(n-1)mⁿ
(10) From Euclid's Lemma, p divides k or w.
(11) It can't divide w since it already divides v and gcd(v,w)=1. Therefore, it divides k
(12) We can apply this same argument for each p in p^(n-1)
(13) So, we can conclude that p^(n-1) divides k.
(14) So, there exists V such that k = p^(n-1)*V
(15) So, kw = p^(n-1)mⁿ = p^(n-1)*V*w
(16) Dividing p^(n-1) from both sides gives us:
Vw = mⁿ
(17) Now, gcd(V,w)=1 since V is a divisor of v and gcd(v,w) = 1
(18) Likewise, V cannot be an n-power. If it were, then v = pⁿV would make v an n-power which goes against our assumption.
(19) Finally, V is less than v since p^(n-1) > 1.
(20) Thus, we have a contradiction by infinite descent.

QED

We proved that assuming that a relatively prime divisor of an n-power is not itself an n-power means that there must necessarily be a smaller relatively prime divisor that is also not an n-power and so on and so on.

Here is what Fermat wrote about Infinite Descent in a letter to another mathematician:

"As ordinary methods, such as are found in books, are inadequate
to proving such difficult propositions, I discovered at last
a most singular method...which I called the infinite descent.
At first I used it to prove only negative assertions, such as
'There is no right angled triangle in numbers whose area is a
square'... To apply it to affirmative questions is much harder,
so when I had to prove 'Every prime of the form 4n+1 is a sum
of two squares" I found myself in a sorry plight (en belle
peine). But at last such questions proved amenable to my methods."
-Quoted from Andre Weil's Number Theory