Thursday, May 19, 2005

Leonhard Euler

The next mathematician in the story of Fermat's Last Theorem is Leonhard Euler, whose name is pronounced "Oiler". Euler was one of the most prolific mathematicians of all time. One of those interests was number theory and he was the first mathematician to make progress on Fermat's famous problem (Fermat, himself, provided a proof for n=4). Euler was responsible for finding a proof for n=3 and for finding a method of proof using imaginary numbers. I will go into detail on both of these points in future blogs.

Euler was born on April 15, 1707 in Basel Switzerland. His father was a Protestant minister with an interest in mathematics. In fact, his father attended lectures by the very famous Jakob Bernoulli. For those interested, the Bernoulli family is one of the most famous families in all the history of mathematics.

Euler entered the University of Basel when he was 14. He originally planned to study theology but his abilities in mathematics attracted the attention of Daniel and Nikolaus Bernoulli, sons of Jakob. The Bernoullis convinced Euler's father to let him study mathematics.

In 1727, he received second place in Grand Prize of the Paris Academy.

At 23, Euler became a professor of physics in St. Petersburg. In 1733, when he was 26, he married Katharina Gsell, the daughter of a Russian painter. They would have 13 children, only 5 of which survived infancy. Euler would later say that he made some of his greatest mathematical discoveries while he had a baby in his hands.

By 1740, he had lost the use of one of his eyes and was starting to lose sight in his other eye. By this time, he had a very strong reputation in mathematics. He had won the Grand Prize of the Paris Academy in 1738 and 1740.

In 1741, at the age of 34, he became Director of the Prussian Academy of Sciences in Berlin. He did not get along well with Frederick the Great and returned in 1766 to St. Petersburg where he stayed for the rest of his life. During this time, he wrote 380 articles, wrote books on the calculus of variations, the mathematics of planetary motions, on artillery and ballistics, on analysis, shipbuilding, and navigation among others. Frederick was apparently very upset when Euler left.

Euler's achievements are legendary. His total mathematical output fills over 70 large volumes. He was able to do all this even while he gradually lost his eyesight. For the last seventeen years of his life, he was completely blind. Almost half of all his output was accomplished during this time. He died in 1783 at the age of 76. Even after his death, it took 50 years to release his unpublished works.

It is really impossible to summarize Euler's achievements in the space of this blog. Instead, I will present just a glimpse of some of his very interesting results.

  • Euler invented the notation ƒ(x) for function (1734), e for natural logs (1727), i as the square root of -1 (1777), the notation π for pi, the notation Σ for summation (1755), and the notation for finite diferences ∇y and 2 y.
  • He was the first to apply Newton's calculus to physics.
  • He showed the usefulness of using imaginary numbers as exponents. Which includes one of the most amazing results in mathematics, known as Euler's identity: e=-1.

I go into detail about this equation here. It is used in the derivation of cyclotomic inteogers which are the basis for Ernst Kummer's proof for Fermat's Last Theorem.
  • He solved many long standing mathematical problems including the Basel Problem, Fermat's Last Theorem for n=3, and many other of Fermat's unproved theorems and conjectures.
  • He made many breakthroughs in physics deriving what are known today as Euler equations which are a set of laws of motion in fluid dynamics that are derived directly from Newton's laws of motions.
  • Along wtih Daniel Bernoulli, he established the law of torque on a thin elastic beam.
In summary, Euler made significant contributions to number theory, differential geometry, mechanics, theory of music, analysis of planetary orbits, differential calculus, lunar theory, differential equations, analytic mechanics, Fourier series, complex numbers, analytic functions of a cmplex variable, logarithm of complex numbers, ordinary and partial differential equations, calculus of variations, mathematical physics, Bessel functions, topology, hydrostatics, fluid mechanics, astronomy, and many other areas which I have not listed.

In 1976, his image appeared on the Swiss 10 franc note.

The contents of this blog were based on two sources:

2. MacTutor Biography of Euler

Tuesday, May 17, 2005

Fermat's Last Theorem: n = 4

The easiest proof for Fermat's Last Theorem is the case n = 4. The proof for n=3 is a bit tricker. What is nice about this proof is that it arises quite naturally from the solution to Pythagorean Triples but it also proves Fermat's Theorem for all values n > 2 where n is even if we can prove Fermat's Last Theorem for all values n > 2 where n is odd. In fact, we can use the same time of reasoning to prove for all values n > 2 where n is not a prime if we can prove it true for all values n > 2 where n is prime.

Many textbooks say Fermat himself published this proof. This is not completely true. Fermat published a proof showing that a right triangle cannot have its area equal to a square. Fermat's proof does by implication show that there is no solution to n=4 but it is a bit more complicated than the proof that I am about to show.

Theorem for FLT/n = 4: There are no integer solutions to:
x4 + y4 = z2 where xyz ≠ 0

(1) We can assume that x2,y2,z are coprime. [From here since (x2)2 + (y2)2 = z2]

(2) From the solution to Pythagorean Triples, we know that there exist p,q such that:
x2 = 2pq
y2 = p2 - q2
z = p2 + q2

(3) Now from this, we have another Pythagorean Triple since y2 + q2 = p2.

(4) So, there exist a,b such that:
q = 2ab
y = a2 - b2
p = a2 + b2
a,b are relatively prime.

(5) Combining equations, we have:
x2 = 2pq = 2(a2 + b2)(2ab) = 4(ab)(a2 + b2)

(6) Since ab and a2 + b2 are relatively prime, we know that they are both squares. [See here for the proof.]

(7) So, there exists P such that P2 = a2 + b2.

(8) But now we have reached infinite descent since:
P2 = a2 + b2 = p which is less than p2 + q2 = z which is less than z2.

NOTE: if xyz=0, then this argument does not hold. This proof only works for xyz ≠ 0.

(9) So, the existence of a solution to the initial equation leads necessarily to the existence of another smaller square that has the same properties.


The case for n=4 is then stated as a corollary.

Corollary for FLT n = 4: There is no solution to:
x4 + y4 = z4 where xyz ≠ 0

Since this equation is equal to:
x4 + y4 = (z2)2

Corollary for FLT n divisible by 4: There is no solution to:
x4n' + y4n' = z4n' where xyz ≠ 0 and n' = (n/4)

Since this equation is equal to:
(xn')4 + (yn')4 = (z(2n'))2

Corollary for FLT n > 2: FLT is proven if FLT is proven for all cases where n is a prime.

The lesson learned from all this, is that we only need to proof that Fermat's Last Theorem is true for prime values of n.

If we prove that Fermat's Last Theorem is true for a given prime number, then it follows that it is true for any number which is divisible by that prime. For example, if we prove that there is no integer solution for x3 + y3 = z3, then we have likewise proven that there is no solution for (xi)3 + (yi)3 = (zi)3 = x3i + y3i = z3i.

This would also prove that x9 + y9 = z9 has no nontrivial integer solution.