Thursday, October 06, 2005

Johann Dirichlet

Johann Peter Gustav Lejeune Dirichlet was born on February 13, 1805 in Duren which at the time was part of Napoleon's empire. His family had originally come from Richelet, Belgium which is why his family name was Dirichlet which means 'from Richelet.'

From an early age, he showed a fascination with mathematics. By the age of 12 when he started at the Gymnasium in Bonn, he would routinely spend his pocket money on math books. At school, he was a model student and after two years, transfered to the Jesuit College in Cologne. By the time he graduated, he decided to seek an education in Paris. There, he was able to attend lectures by some of most famous mathematicians of the time including Fourier, Laplace, Legendre, and Poisson.

In 1823, he began to work for the family of General Maximilien Sébastien Foy. Dirichlet taught German to Foy's wife and children. Dirichlet chose at this time to work on his first paper which concerned Fermat's Last Theorem for n=5. At this time, case for n=3 and n=4 had already been solved. In 1825, Dirichlet succeeded in proving it true for the case where one of the numbers x,y,z is divisible by 10. The paper attracted a lot of attention and one of his reviewers was Adrien Legendre. Shortly after the presentation Legendre was able to complete the proof for n=5. Dirichlet also came up with a complete proof but did so only after Legendre had published his complete solution for n=5. In addition, he wrote a very important paper on biquadratic reciprocity which extended a result by Gauss.

Around this time, General Foy died and Dirichlet returned to Germany. He sought employment but was denied because of the requirement that he needed to have completed a doctoral thesis. This problem was solved when he was given an honorary doctorate from the University of Cologne. He then acquired a post at the University of Breslau. Despite his fame from his result with Fermat's Last Theorem, his appointment at the University of Breslau created a great controversy among the German math professors.

Dirichlet found the standards at the university were disappointingly low and in 1828, he transferred to the University of Berlin. He remained there until 1855. Later, he would help to greatly raise the mathematical standards in Germany.

In 1831, he was appointed to the Berlin Academy. By this time, he was paid enough to consider marriage. He married Rebecca Mendelsohn who was a sister to the composer Felix Mendelsohn. At this time, he also corresponded with the mathematician Jacobi. Together, their correspondences exerted a tremendous influence on number theory.

In 1837, he proved a conjecture by Gauss on the arithmetic progression of primes that clarified a result done earlier by Legendre. His work would later form the foundation of analytic number theory and algebraic number theory. That same year he proposed what is today the modern definition of a function. Other work included potential theory, integration of hydrodynamic equations, convergence of trigonometric series, and fourier series.

In 1855, after Gauss's death, Dirichlet took over Gauss's post in Gottingen. He was very happy in Gottingen but unfortunately, his stay there did not last long. In 1858 while lecturing at a conference in Montreux, he suffered a near fatal heart attack. When he returned to Gottingen, he learned that his wife had just died of a stroke. He died shortly afterwards in May of 1859.


Wednesday, October 05, 2005

Adrien-Marie Legendre

Adrien-Marie Legendre did not like to talk about his personal life. His colleague Simeon Poisson wrote:
Our colleague has often expressed the desire that, in speaking of him, it would only be the matter of his works, which are, in fact, his entire life. (From MacTutor Biography)
He was born September 15, 1752 in France. He may have been born in Toulouse but from a very early time, his family lived in Paris. His family was wealthy and he attended the College Mazarin in Paris. His area of focus was mathematics and physics.

In 1775, he became a lecturer at the Ecole Militaire based on the recommendation of the well known mathematician Jean D'Alembert. One of his fellow lecturers at the Ecole Militaire was Pierre-Simon Laplace who would later become very well known himself.

Legendre's reputation was made when he was able to win the Berlin Academy Prize in 1782. The Berlin Academy had proposed a very difficult problem that involved calculating the projectory path of a cannon ball undergoing air resistance. He later wrote an influential paper on the force of attraction between two confocal ellipsoids. In January of 1783, he was appointed as an associate to the French Academy of Sciences.

Legendre proceeded to do influential investigations of celestrial mechanics, number theory, and elliptic functions. In number theory, his work includes an important result on the quadratic reciprocity of residues and arithmetic progressions of prime numbers. Both of these results were later superceded by Carl Friedrich Gauss's work on quadratic reciprocity of residues and Johann Dirichlet's work on arithmetic progressions of primes. In 1787, he was elected to the Royal Society of London.

After the French Revolution of 1793, Legendre lost most of his fortune. In 1794, he published his Elements of Geometry which became the leading elementary text on geometry at the time. In this work, he attempted to make the proofs behind Euclid's Elements clearer and better organized.

In 1801, Gauss criticized Legendre's results on quadratic reciprocity and claimed that he himself had invented the correct method. It is clear that Legendre was not happy with Gauss's words but in 1808, when Legendre came out with the next version of his textbook on number theory, he included Gauss's proof instead of his own.

In 1824, Legendre refused to support the government's candidate for the National Institute. As a result of his refusal, he lost his pension and lived his remaining years close to poverty.

In 1830, he offered his proof for Fermat's Last Theorem n = 5 which was based on the work done by Dirichlet and also the proof by Sophie Germain.

Legendre was fascinated by Euclid's parallel postulate and for many years attempted to provide a proof. He refused to believe in the existence of non-Euclidean geometries which were first proposed by Nikolai Ivanovich Lobechevsky in 1829.

Legendre died on January 10, 1833. He had made significant achievements in geometry, differential equations, calculus, function theory, number theory, and statistics.


Monday, October 03, 2005

Continued Fractions

The proof for Fermat's Last Theorem n = 5 depends on Continued Fractions. A Continued Fraction is an expression of the following form:

where a0, a1, etc. are all integers. To simplify this characterization, the Continued Fraction is represented using the following notation:
[ a0, a1, a2, ... ]

To understand the power of continued fractions, let's look at how they can be used to represent the square root of 2. Being an irrational number, the digits that make up the real number do not repeat (if they repeated, then it would not be an irrational number, see here). The digits we get are: 1.4142135...

Using continued fractions, we can represent this same value as [1,2,2,2...] where 2 repeats for ever (see below).

Continued Fractions as an idea have their origin withEuclid's Algorithm for finding the greatest common denominator of two integers. Popular study of Continued Fractions began with the work of John Wallis who was the first to use the term "continued fractions."

Later, Leonhard Euler established the foundations of continued fractions. He demonstrated, for example, that every rational number can be expressed as a terminating, simple continued fraction, established a formula for the constant e in continued fraction form, and used this to prove that e is irrational (more on this in a future blog)

Other important work was done by Joseph Louis Lagrange. He used continued fractions to show the value of irrational roots and proved that the real root of a quadratic irrational is a periodic continued fraction (more on this result in a future blog).

The story of continued fractions goes greatly beyond this short outline. For those interested in understanding the larger story, please check out the references at the bottom of this blog.

First, it should be clear that all real numbers can be represented as continued fractions.

Lemma 1: We can generated a continued fractions of the form [ a0, a1, a2, ... ] for any real number.

(1) Let α be any real number.
(2) Let's define a function floor() such that the floor() of any real number is the highest integer that is lower than the real number. For example floor(π) = 3.
(3) Now, here are the rules for constructing the continued fraction:
(i) Let α0 = α
(ii)Let an = floorn)
(iii) Let αn+1 = 1/(αn - an)
(4) For any integer, the continued fraction is [a0] since floor(any integer) = the integer.
(5) If it is not an integer, then we use (iii) to get α1 and use (ii) to determine a1.
(6) In this way, we can generate a continued fraction for any real value.


As an example, let's use the algorithm above to generate the continued fraction for 2:

Corrolary: √2 = [ 1,2,2,2...]

(1) a0 = floor(√2) = 1.

(2) α1 = 1/(2 - 1) =2 + 1 [By multiplying both sides by 2 + 1 ]

(3) a1 = floor(√2 + 1) = floor(√2) + 1 = 2

(4) α2 = 1/(√2 + 1 - 2) = 1/(√2 - 1) = √2 + 1.

(5) This means that we are at the same value as step (2) which means that the value 2 goes on ad infinitum.