What is Number Theory?
1.
Historical Background
Number theory is one of the oldest branches of mathematics, dating back to pre-Socratic
Greek thinkers such as Pythagoras and his school (ca. 550 B.C.) who knew how to solve the
famous “Pythagorean” equation x2 + y2 = z2. Moreover, recent cuneiform evidence suggests that
mathematicians from ancient Babylonia knew about this equation as far back as 1600 B.C.
(more than 3600 years ago!) In fact, it appears the Babylonians were capable of computing all
integral solutions x, y, z of this equation and used this knowledge to construct crude
trigonometric tables.
Subsequent to Pythagoras, famous Greek mathematicians such as Euclid (ca. 300 B.C.) and
Diophantos from Alexandria (3rd century A.D.) were enormously influential in the development
of number theory. Their work was continued by mathematicians in medieval Arabia and Italy,
such as Al-Karkhi (ca. 1030) and Leonardo Pisano (ca. 1200).
In the modern era, influential figures in number theory include Pierre de Fermat (French, 17th
century), the founding father of the discipline, Leonhard Euler (Swiss, 18th century), Carl
Friedrich Gauss (German, early 19th century), Jean de la Vallée-Poussin (Belgian, late 19th
century), G. H. Hardy (British, early 20th century), Paul Erdös (Hungarian, mid 20th century) and
Andrew Wiles (British, late 20th century). This is by no means an exhaustive list, but simply a
few of the most famous names associated with the development of number theory.
At this moment, some of the most important open questions in mathematics come from
number theory, in particular the Riemann Hypothesis which is related to the distribution of the
prime numbers (the Clay Institute in Cambridge, MA has pledged to give $1 million to anyone
who can solve this fundamental question). Other famous number-theoretical questions include
Goldbach’s Conjecture, the Twin-Prime Conjecture and Fermat’s Last Theorem.
2.
The Queen of Mathematics
Carl Friedrich Gauss, who pioneered virtually every field of mathematics in the early 19th
century (he is sometimes called the “Mozart of math” given the incredible range and brilliance
of his work), described number theory as the “queen of mathematics.”
His position is similar to that of most mathematicians who have always regarded number theory
as the purest form of mathematics.
One of the reasons for such high praise is that very simple questions in number theory can lead
to extraordinarily difficult answers (or, sometimes, to no answers at all!) A striking example of
this is Fermat’s Last Theorem, which states that the equation xn + yn = zn has no integral
solutions x, y, z for any natural number n greater than 2. This is actually a generalized version
of the famous Pythagorean Theorem, but with a negative answer!
In 1647, Fermat claimed that he had a marvelous simple proof of this important result.
Unfortunately for him, no one has ever found a trace of it. It eventually took more than 350
years – and countless failed individual efforts by some of the finest minds – before Andrew
Wiles, working at Princeton University, could prove it and put the matter to rest. Wiles first
read the problem as a 10-year-old in a math book. Later, as an adult, he became fascinated by
it and spend years in complete seclusion trying to solve the puzzle. He became an instant
worldwide sensation in 1993 when he finally emerged with a proof*.
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*Wiles’ original proof turned out to have a major flaw in it. This took months to detect due to the difficulty that
referees had in going over such a technically- advanced proof. Despite this setback, Wiles went back to work on
his proof and found a way to get around this flaw. He resubmitted a proof in 1994, which was eventually validated
by his peers in 1995.
3.
A Higher Arithmetic
So we are back to our initial question, “What is number theory?”
A simple answer is that number theory is primarily concerned with the properties of the
natural numbers, also called the counting numbers (1, 2, 3, 4, 5, …). By extension, number
theorists also study the properties of other number sets, in particular the whole numbers (0, 1,
2, 3, 4, …), the integers (…, -3, -2, -1, 0, 1, 2, 3, …), and the fractions.
You might think at this point that we are taking a step back to basic arithmetic, but that is
actually not the case at all! In fact, number theory is commonly referred to as “a higher
arithmetic.” To understand this further, think about what you really know about a natural
number like, say, 31. Unless you use this number in a sum, a product, a ratio, or any other
arithmetical computation (for example, totaling a grocery bill where one of the item you bought
cost $31), you might not think twice about the natural 31. Yet, this number is interesting in its
own right. For one thing, it is prime because its only factors are 1 and itself. Furthermore, it
also has a twin prime: 29. The number 31 is also deficient. And so on.
The point here is that number theory is ultimately concerned with finding all the interesting
properties that apply to the natural numbers. This is quite different from elementary
arithmetic, which merely uses these naturals for the purposes of computing sums, differences,
products, ratios, etc.
So what actually constitutes an “interesting” property?
The following famous anecdote will illustrate what could be considered “interesting” to a
number theorist. In the early 20th century, the British number theorist G. H. Hardy went to
visit his young protégé, an Indian named Srinivasa Ramanujan, who was sick in the hospital.
Ramanujan was uneducated mathematically (he came from a very poor background), but
possessed a remarkable insight into hidden arithmetical relationships. For that reason, Hardy
had asked him to come to England and work with him. When he arrived to see him at the
hospital, Hardy remarked that the taxi in which he had ridden had the license number 1729,
which, he said, seemed a rather uninteresting natural. Quickly Ramanujan replied that, on the
contrary, 1729 was singularly interesting. In fact, he said, 1729 is the smallest natural that can
be expressed as a sum of two cubes in two different ways:
1729 = 103 + 93
and
1729 = 123 + 13
What should be inferred from this anecdote is not that one should be capable of doing lightning
fast computations like Ramanujan to be a successful number theorist; or that one needs to
know obscure facts about random numbers (such as 1729) to understand number theory.
Instead, the pertinent point here is that Ramanujan’s remark can lead to all sorts of interesting
questions whose answers are by no means simple matters of calculation. For example, here’s
an interesting question that could be posed based on Ramanujan’s observation: “If s is a natural
greater than 2, how large is the smallest natural that can be represented as a sum of cubes in s
different ways?” Or another simpler question could be: “Are there any cubes that can be
expressed as the sum of two cubes?”
The first question we asked is very hard to answer. The last question, however, is a special
case of Fermat’s Last theorem since we are asked to find integral solutions x, y, z of the
equation x3 + y3 = z3. We now know – thanks to Wiles – that the answer here is a resounding
“No.”
4.
Number Theory in the 21st Century
Nowadays, number theory plays a major role in our way of life, as illustrated by the three
examples below.
The encryption systems that are used by internet websites such as ebay.com, amazon.com,
and countless others to secure transactions on their sites operate solely on numbertheoretical principles.
ISBN, UPC and credit card numbers are all established based on certain congruences – a
major topic in number theory. This is done in order to minimize errors in transcription.
The National Security Agency, a critical American agency in charge of national security, uses
sophisticated tools in cryptography (i.e. the practice and study of mathematical techniques
used for securing communications between two parties in the presence of a third party,
called the “adversary”) to secure the U.S. and its allies from external threats such as cyberterrorism. This is all done, in large part, thanks to the work of hundreds of number
theorists within the intelligence community.