Pythagorean Triples: Solution

OK, here's the solution for Diophantus's problem of determining the solution for:
x² + y² = z²

(1) We know that we can assume that x,y,z are coprime. [See my previous blog for details]

(2) The second important insight is that z has to be odd.

(a) Assume the opposite that z is even.

(b) Then, there exists another value Z such that z = 2 * Z

(c) Also, z² is then divisible by 4 since:

z² = (2 * Z)² = 4 * (Z²)

(d) We know that x,y must both be odd because of (1).

(e) Since they are odd, there must also exist values X,Y such that:

x = 2 * X + 1
y = 2 * Y + 1

(f) But x² + y² cannot be divisible by 4 since:

x² + y² = (2 * X + 1)² + (2 * Y + 1)² =
= 4X² + 4X + 1 + 4Y² + 4Y + 1 =
= 4[ X² + X + Y² + Y ] + 2

(g) So, we have a contradiction and we reject our assumption.

(3) Since z is odd, either x or y must be even since an odd number is always the sum of an odd and an even number.

(4) Let's assume x is even. The same argument will also work if y is even.

(5) Now, we know that:

x² = z² - y² = (z - y)(z + y)

(6) And, z - y and z + y must be even since z,y are odd.

(7) So, we know that there must exist u,v,w such that:
x = (2u)
z + y = (2v)
z - y = (2w)

(8) Which means that:
(2u)² = (2v)(2w) [From (5) and (7)]

(9) Dividing both sides by 4 gives us:
u² = v * w

(10) We need 1 more insight before the solution. Here it is: v,w are coprime

(a) Assume that v,w are not coprime.

(b) Then, there exists d such that d > 1 and d divides both v,w

(c) Then d divides both v + w and v - w

(d) But:

z + y + z - y =
2v + 2w
So 2z = 2v + 2w which means that z = v + w
So d divides z

(e) And:

z + y - (z - y) = 2v - 2w
So 2y = 2v - 2w which means that y = v - w
So d divides y

(f) Which is a contradiction since z,y are coprime [by (1)].

(g) So, we reject our assumption.

(11) By the properties of coprimes, we know from (9),(10) that v,w are themselves squares (see here for proof). [For those who need a review of coprimes, here is a link.]

(12) So, there exists p,q such that:

v = p²
w = q²

(13) And, we have our solution since:

z = v + w = p² + q²
y = v - w = p² - q²
x = 2u = 2pq [Since u² = vw means u = pq]

We also know that:

(a) p,q are relatively prime. [Otherwise, z,x,y would not be relatively prime]

(b) p,q are opposite parity (that is, one is odd and one is even) [Since z is odd]

(14) For sure enough:

(p² + q²)² = (2pq)² + (p² - q²)²

(15) Now, to generate our answer, we can pick any p,q we want so long as they are integers.

For example, if p = 2 and q = 1, we get:

z = (2)² + (1)² = 5.
y = (2)² - (1)² = 3.
x = 2pq = 2*2*1 = 4

(16) We can do even better than this because we know that for each x,y,z, if they have common factors, the relation still holds.

z = d[p² + q²]
y = d[p² - q²]
x = d[2pq]

For example, if p = 2 and q = 1 and d = 2, we get:
z = (2)(5) = 10
y = (2)(3) = 6
x = (2)(4) = 8

And sure enough, 6² + 8² = 36 + 64 = 100.

QED

What's also nice about this result is that it is not too difficult to apply Fermat's method of infinite descent and prove Fermat's Last Theorem for n=4.

Coprime Numbers

In the last blog, I spoke about Diophantus's problem: to divide a square into the sum of two smaller squares.

In other words, to find solutions for x,y,z where:
x² + y² = z².

The first step in solving this problem is to realize that we can assume that x,y,z are coprime (or another way to say it, relatively prime). That is, no two of these values are divisible by the same prime. So, if p is a prime that is a factor of x, then we know that it is not a factor of y and not a factor of z.

When we have a situation where the three numbers are not coprime (for example, 6,8,10), we will be able to divide out common factors and end up with three numbers that are.

In the case of 6,8,10, the three numbers share the prime 2. If we divide out 2, then we are left with 3,4,5 which are coprime.

This assumption is important because it greatly simplifies the task of analyzing the conditions for when a solution exists. In my next blog, I will show how this assumption gives us the solution to Diophantus's problem.

Interestingly, we can apply this same assumption to Fermat's Last Theorem. From this point on, we will only need to consider the case where x,y,z are relatively prime.

One of my goals in this project is to provide complete proofs each of the conclusions presented. This blog relies on one lemma. A lemma is an intermediate statement that requires proof and is used in a larger theorem.

Lemma: All solutions to xⁿ + yⁿ = zⁿ can be reduced to a form where x,y,z are coprime. [Here is the proof.]

Pythagorean Triples

When Fermat wrote his note in the margin, he was making a comment on the problem of determining Pythagorean Triples.

In Book II, Problem 8 of the Arithmetica, Diophantus poses the problem of how to divide a given square number into the sum of two smaller squares.

In other words, solve the problem:
x² + y² = z².

Any three numbers that satisfy this equation are called Pythagorean Triples. They are called Pythagorean Triples since this is the same equation as the Pythagorean Theorem.

The Pythagorean Theorem is so well known that I refer people to this link if you would like to see a proof for it.

An example of a Pythagorean Triple is 3, 4, and 5 since 3² + 4² = 5².

I encourage everyone who has not already seen the solution to Diophantus's problem to try and solve it. This is without doubt what Fermat did and in solving this problem, he stumbled upon his famous generalization.

If you solve the problem, you should be able to prove there an infinite number of Pythagorean Triples and find a method for listing them out.

You can find the solution here.

This blog is based on the following sources:

• Timeline of Fermat's Last Theorem

Diophantus of Alexandria

When Fermat made his famous note in the margin, he was making a comment on a problem from Diophantus of Alexandria. While Fermat is today considered the father of number theory, he would probably have given this title to Diophantus.

Not much is known about Diophantus's life. He was a Hellenized Babylonian who lived in Alexandria, Egypt. He was born around 200 AD and he died around 284 AD.

His classic work is the Arithmetica which consists of 130 problems in 13 books. Today, only 6 of the books survive. It was written around 240 AD.

He focused on problems that have a positive rational number as a solution (a rational number is any fraction made from two whole numbers). He considered negative numbers and irrational numbers to be "useless," "meaningless," and "absurd."

When he spoke about quadratic equations, he only offered one solution. It is not clear if he knew that all quadratic equations have two solutions.

Diophantus is considered to be the father of algebra. Diophantine equations are named after him.

Much of what we know about him comes from the words written on his tombstone, which is itself an algebra problem for determining his age:

This tomb hold Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers forfour years, he reached the end of his life.

This blog is based on the following sources:

• The Biographies of Mathematicians Web Site
• J.R. Newman, The World of Mathematics
• Wikipedia

Fermat's Achievements

Pierre de Fermat published very little in his lifetime, and yet by most accounts, he is considered to be one of the greatest mathematicians of all time.

I was very surprised when I learned that Fermat's Last Theorem was published after Fermat's death. It is pretty amazing that the mathematical world took a note in the margin so seriously.

Fermat was very precise about his mathematical pronouncements. When he was not certain about a finding, he labeled it a conjecture. Indeed, shortly after Fermat's death, all but one of his theorems had a proof just as Fermat had claimed.

Fermat was born in Toulouse, France on August 17, 1601. He was born into a wealthy family. He studied law at the university in Orleans and became a councillor at the parliament in 1631. He quickly moved up the ranks and in 1652, he became the chief magistrate of the criminal court.

In 17th century France, magistrates were required to spend large amounts of time in isolation. It was during this time, that he worked on mathematics.

His achievements in mathematics are incredible.

Sir Isaac Newton said that his invention of calculus was based a large part on Fermat's method of tangents.

In 1654, Blaise Pascal wrote a letter asking about Fermat's views on probability. Their series of correspondences became the foundation of probability theory.

Rene Descartes is perhaps most famous for his invention of Cartesian coordinates and his classic work La Geometrie. Fermat independently came up with a three-dimensional geometry. While Fermat's version of analytic geometry was more complicated and advanced, Descarte's classic work became more popular because its notation was more convenient. Today, both men are seen as the fathers of analytic geometry.

He made advances in optics. He came up with Fermat's Principle, the idea that light always traverses the path that takes the least time.

With Wiles' amazing proof, we are still left with a question. Did Fermat really have a proof?

Wiles' proof rests on twentieth century mathematics including the theories of elliptic curves, modular forms, and Galois Representations.

Wile's proof goes way beyond the mathematics that Fermat helped to create. It is very possible that Fermat's proposed proof made an incorrect assumption about cyclotomic factorization. This was a mistake that got Gabriel Lame into falsely believing that he had found a proof of Fermat's famous problem (I will be talking more about this in a later blog).

It is also possible that Fermat, in his lifetime, figured out that he had made a mistake. If he had truly found such a marvelous proof, why did he never tell anyone about it while he was alive? Perhaps, he had figured that he had made a mistake but forgot to correct his notes.

Still, the possibility that Fermat had a proof raises an interesting question: is there a way to simplify Wiles' result? Is it possible to derive a result from Wiles' proof that corresponds to the mathematics of Fermat?

The information in this blog was taken from the following sources:

• MacTutor
• Simon Singh's Web Site (author of Fermat's Enigma)
• Wikipedia

The Problem

For this blog, I assume that everyone is familiar with exponents. If you are not, here is an introduction.

Fermat's Last Theorem states for the equation: xⁿ + yⁿ= zⁿ, there are no whole number solutions where x * y * z ≠ 0 and n > 2.

If x * y * z = 0, then it is easy to find a solution. For example (5)ⁿ + (0)ⁿ = (5)ⁿ.

Likewise, if we consider real numbers, then the solution is straight-forward algebra:
z = (xⁿ + yⁿ)^(1/n).

Finally, if n = 2, then we have the Pythagorean Theorem a² + b² = c². This is solveable by any Pythagorean Triple such as 3,4,5 (3² + 4² = 5²) .

I think that this is the real appeal of the problem. It is easily stated and on its surface looks like it shouldn't be too difficult to resolve one way or the other.

Pierre de Fermat rarely published any of his results. He prefered to describe the problem and claim that he had found a solution. This has made the problem even more appealing: did Fermat actually have a proof?

The theorem itself became public without proof in 1670 when Fermat's son, Clement-Samuel published his father's notes. Unfortunately, Fermat was not around to explain his famous theorem because he had died in 1665. Instead, the reader was left with the famous statement of the problem:

"It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum oftwo like powers."

And this very mysterious statement about the proof:

"I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain." (both quotes are from Fermat's Engima)

For over 350 years, this problem remained unsolved. Many of the greatest mathematicians were able to make progress on the problem including Leonhard Euler, Carl Friedrich Gauss, and Ernst Kummer but none of these great minds offered a solution.

The solution had to wait until 1995.