22
Brahmagupta & the Linear Equation
To modern eyes, it may have seemed curious to not have started the course with the
simplest of all equations, the linear equation ax by c . The fact is that rational
solutions to such an equation are trivially found by
123
Lagrange & Primitive Elements
The greatest French mathematician of the 18
th
century was Joseph-Louis Lagrange
(1736-1813). He was so capable that many consider him Eulers equal. He contributed in
the French Revolution and participated in the creation
79
Fermat & his Little Theorem
We have seen that our subject has a long history and tradition in many cultures and
countries. But if we consider our culture as part of the culture born out of the
Renaissance, then one of the people most responsible for br
94
Euler & His Bigger Theorem
The best mathematician of the eighteenth century was the ubiquitous, omnipotent Euler,
who not only was the first one to prove Fermats Little Theorem (it was his proof of this
fact that we gave in the last section), but he al
109
Dirichlet & Multiplicative Functions
When the great Gauss passed away, his position at the University of Gottingen was
occupied by Lejeune Dirichlet (1805-1859), who intellectually had been close to Gauss
all his mathematical life. Dirichlet proved a
163
Riemann & the Distribution of Primes
Soon
after Fermats Last Theorem was solved in the early 1990s (which was
mentioned in the first section), it was replaced as the most famous unsolved problem in
mathematics by the Riemann Hypothesis. Georg Riemann
66
Qin Jiushao and Many Clocks
So far we have been to Alexandria thrice, and once each to India and 19th Century
Germany. In this section we travel to China, where problems involving more than one
congruence at a time are quite ancient.
If all we have is
53
Eratosthenes & His Sieve
Everybody knows how to multiply two numbersalthough few of us are appreciative
of the many contributors from the past that gave us the wonderful notation and algorithms
that allow us to multiply rather painlessly. Just 600 year
33
Gauss & Modular Arithmetic
The ideas of modular arithmetic are several hundred, if not thousands, years old. The
Vedic transformation, for example, was an ancient Indian tradition, which consisted of
adding the digits of a number, and doing it again un
10
Euclid and the Reduction of Fractions
In the previous section we already encountered the notion of a reduced fraction. The
equivalent notion is that of two relatively prime integersin other words, a fraction is
reduced if and only if its numerator and
1
Diophantus and the Search for Rational Solutions to Equations
Number theory is our subject and it concerns properties of what some consider the only
true numbers, the natural numbers, , together with their negatives, the integers,
and with their quotien
139
Legendre & his Symbol
The French Revolution had among its leaders and followers several distinguished
mathematicians. Of course, we have already met the great Lagrange and most have heard
of Laplace, a leader in differential equations, probability an