Sunday, October 31, 2010

Fermat's Last Theorem - A Primer

Fermat's Enigma, written by Simon Singh, is a book about Fermat's Last Theorem, its origin, history and how it was officially proved in 1997. One day Lily picked up this book for kids' non-fiction reading. I scaned a few pages and could not put it down - I was fascinated.

Fermat's Last Theorem

The theorem was in fact a conjectecture, first proposed by french mathematician Pierre de Fermat, around 1637. It was based on the Pythagorean Theorem:

In a right hand triangle, the square of the hypotenuse (z) is equal to the sum of the square of the other tow sides, i..e. x^2+y^2=z^2.

A peculiar feature of the equation is that it has infinite whole number (integer) solutions, e.g. (3,4,5), (5, 12, 13), .....

It is noted that Chinese and Babylonians discovered the relation one thousand years before Pythogoras. However it was mathematically proved by Pyththagoras of Samos, an ancient greek mathmatician (~ 600BC), and thus the name after him.

Fermat's last theorem
The following equation

x^n+y^n=z^n (*)

has no integer solution if n>2.

It is intriguing that equation (*) has infinite number of integer solutions for n=2, and has no integer solutions at all for n>2. For 350 years since the proposal of the theorem no body can prove or disapprove the theorem.

People behind the Math

The math involved in the history of proving the Fermat's last theorem is interesting. The life stories of the involved mathematicians are fascinating - their life, their work, their failures in proof, triumphs in reaching major milestones. Five mathematicians were of great significance to the Fermat's last theorem and its proof: Pythagoras(600BC), whose Pythagorean theorem inspired P. Fermat to make the conjecture; Fermat, who intrigued and frustrated mathematicians centuries to follow with his notes on the margin of a mathematics book(1637); L. Euler, the 18th century genius, who made the first breakthrough in proving Fermat's last theorem (1753), 100 years after Fermat's death; P. Wolfskehl, who helped to sustain interests in the proof of the theorem by setting up a Wolfskehl Prize in 1908 and Andrew Wiles, the Princeton professor who proved Fermat's last theorem in 1997.

Fermat, who was actually a judge, was brilliant and smarter than most of his contemporary mathematicians. He was into math because it was fun for him. He did not publish any of his work because he did not like to provide full proof of the problems he solved. He usually just wrote down an outline of how he solved the problem. Of course the most famous such sketch is the note he left on the margins of the book II of Arithmetica. He stated his conjecture there and then he continued to write " ...I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain". However he did enjoying challenging others by posting problems he solved. He had particular pleasures to see others struggle on those problems! Thanks to Fermat's son Clement-Samuel, who collected his father's notes and published them in the book Arithmetica Containing Observations by P. de Fermat, we know Fermat's remarkable work in number theory, and the ensuing centuries of struggle to prove his last theorem.

Pythagoras of Samos, an ancient Greek mathematician (~ 600BC) traveled the world to learn math, especially from Egyptians and Babylonians. He laid the very foundation of mathematical proof on top of Pythagorean theorem. A very interesting story was that he paid his first student three silver coins for each lesson taken. After sometime, the boy became enthusiastic about mathematics. Pythagoras then stopped paying his student, pretending that he did not have any more money. The student offered to pay to continue to receive education from Pythagoras.

To advance mathematics, he established Pythagoras Brotherhood, a cult type of mathematical school. He believed everything was a rational number - the whole numbers and fractions. He took this belief so sacredly that it prevented him from accepting the existence of irrational numbers, such as square root of 2. A student of Pythagoras, Hippausas attempted to find a fraction for square root of 2, and eventually he proved that no such fraction existed. Pythagoras did not want to admit the existence of irrational number, but he could not deny its existence by mathematical logic. Tragically he resorted to force, sentencing the student to death by drowning!!

Leonhard Euler, the great mathematician from Swiss, worked in Russia most of his life. Contrary to Fermat, Euler liked to write papers. He proved the theorem for n=3 using imaginary number to plug holes in his proof. His attempts to extend the proof all ended in failure. Euler showed exceptional talent in math at young age but his father wanted him to study theology, and he dutifully obeyed. Luckily the Bernoulli family - one of the Bernoulli was responsible for the Bernoulli principal in fluid mechanics, was in the same town Euler lived and two of the Bernoullis were Euler's good friends. The Bernoullis appealed to Euler Sr. to let Euler pursue a career in math. Because Euler Sr. had great respect to the Bernoullis, he reluctantly gave in. The world got one of its greatest mathematicians.

Paul Wolfskehl was an average mathematician from a wealthy family. He created Wolfskehl Prize due to an aborted suicide. The story goes like this. A failure in his pursuit of a beautiful woman led to his plan to kill himself. He set a date and would kill himself at exactly midnight that day. He was so efficient in carrying out his plan, he finished the preparation for his suicide attempt ahead of time. So he went to the library to scan through mathematical publications and he happened to read a paper on why two major efforts to prove Fermat's last theorem by Lame and Cauchy respectively failed. He was quickly absorbed into the paper and found an error in it. By dawn he remedied the error in the paper, and his despair evaporated. He tore his will and wrote a new one to create a prize for the proof of Fermat's last theorem.

Andrew Wiles, the Princeton math professor, who worked in secret for 7 years during his pursuit of the proof, has a life long obsession with Fermat's last theorem starting at 10 after he read a history book on math. After he announced that he had proved Fermat's last theorem at a math conference, his proof was subject to peer review. Referees found a potential crushing error in the proof. Wiles sweat for about a year and with substantial help from others, finally fixed the error in the proof. The proof was eventually published in the journal Annals of Mathematics in 1995. The Wolfskehl Prize was presented to him in 1997.

(to be continued ... Numbers)

No comments:

Post a Comment