around 3000 ~ 2000 BCE 

In those ancient times Babylonians and Egyptians dealt with elementary number theory for practical reasons such as for solving problems in astronomy, architecture and surveying.  The Babylonians used a number system with basis 60, the Egyptians used a number system with basis 10; neither of them knew or even used a sign for zero which was a later invention.

The Babylonians had lists of reciprocals, multiplication tables and square tables. They used their knowledge of numbers for calculation of interests, transport, civil  engineering and other applications. Both linear equations of up to three unknowns, and quadratic and biquadratic equations were studied.  Areas of triangles and circles were calculated, and the Pythagoras Triangle was known.

The Egyptians used their knowledge of elementary number theory for bills, buskins and calculations of calendars.  They did calculations on rectangles, triangles, trapezoids and circles (with = 4 (⁸/₉)² = 3.16), and were comfortable using fractions. They knew about the golden ratio and used it for harmonious design of buildings. The Pythagorean equation   3² + 4² = 5²  made it possible for them to create right triangles.
 
 

around 1000 ~ 100 BCE 

The Greeks made the first attempts to systematize the disciplines of the natural sciences. There were many Greek mathematicians who made significant contributions to ancient number theory, among them: Thales, Pythagoras, Euclid, and Diophantos. 

The Pythagorean School differed between odd and even numbers, created the concept of divisibility and the concept of perfect and amicable numbers, and knew about the series {0, 1, 3, 6, 10 ... }, the so-called series of triangular numbers, created by adding successing integers to their predecessors. (A perfect number is a number for which the sum of all its factors is twice the number itself. e.g.  6 is a perfect number since the sum of its factors is   1 + 2 + 3 + 6 = 12 = 2 6. Amicable numbers are a pair of numbers whose sum equals to the sum of the factors of each. e.g. {284, 220} are amicable numbers, the sum of the factors respectively being 504.)

Euclid wrote the most famous mathematical treatise of that time: the Elements, containing 13 books, of which only 6 are remained. The books VII – IX are about number theory and contain most significant examples of his works:
 

1. The Euclidean Lemma which states that  p | ab  p | a or p | b   for p prime.

2. The Euclidean Algorithm which finds the greatest common divisor of two numbers which are not prime to one another

3. The Fundamental Theorem of Arithmetic, stating that   ‘any integer can be represented as a product of primes. Such a representation is unique up to the order of prime factors.’

4. Proof of the infinite numbers of primes: suppose p is a prime, consider the number defined as 1 + product of all primes less than or equal to p, i.e.

                            n = 2 3 5 7 11 ... p + 1

             then if n is prime  =>  there is a prime greater than p.
 
 

Euclid	Pythagoras	Pythagorean Triangle
Euclid’s Elements


   

Middle Age and Renaissance

After Euclid the theory of numbers stagnated for almost 400 years during which time the Greeks dedicated their research more to geometry. By the start of the Middle Ages mathematical knowledge had not progressed greatly, but had spread among the Arabs, who made some advances number theory. In Europe many wars and religious fanaticism hindered the pace of science. Among the Arabs, Al-Khowarizmi in the 9th century is especially worth mentioning as a famous scientist in the fields of algebra, number theory and astronomy.

Fibonacci (ca 1170 – 1250) was one of the first to work seriously on mathematics again in Europe. He was a travelling scholar and published the Liber Abaci. He systematically used arabic numbers including the zero and invented negative numbers for debts. Today we connect his name with the recurrent sequence of Fibonacci numbers:
 

x_n+2 = x_n+1 + x_n,     where x₀ = x₁ = 1.


Fibonacci showed in 1228 that
 

there is no integer or fraction or square root of a rational number which satisfies the equation

x³ + 2x² + 10x = 20.
 

This result represented important preparatory work for the solution of cubic equations by radicals, which was succeeded only in the 16th century by Scipione Dal Ferro, Tartaglia and Cardano. Later Cardano’s student Ferrari was able to solve biquadratic equations by radicals.

After Fibonacci number theory was almost stationary. Long wars on the continent absorbed the energy of the people and science was not of any common interest.

Then,  about 400 years later, Viète, a French lawyer who devoted his free time to the geometry of planetary motions, decoding algorithms and algebra, appeared as the next algebraist and number theorist in the mathematical sky. Even though his major contributions were in the fields of trigonometry, he still presented new methods for solving equations of degree 2, 3, and 4. His most important idea on algebra was probably the realization of the connection between positive roots of equations and the coefficients of the different powers of the unknown quantity. Also, he can be regarded as the first to introduce letters as symbols for unknown variables. He began using letters at the beginning of the alphabet for known quantities and letters at the end of the alphabet for unknown quantities, which we still do today, although few of us will recognize this habit as an historical convention.
 
  

Fibonacci

Viète

The frame for mathematics in the 16th and 17th century 

At that time, since there existed no professional scientific societes nor any discipline of mathematics, there was no focus for any scientist towards a mathematical audience. Mathematicians dealt with problems mainly for their own satifaction, and maybe also for the understanding and acknowledgment of their fellow colleagues. Since the attitudes of mathematicians towards mathematics were widely different, it was no suprise that mathematics itself was divided into ‘schools’ which differed from each other not only in the field of research, but also, significantly, in the philosohpical view of mathematical matters and therefore in the method of approach (although there were outstanding mathematicians of that time who belonged to more than one school). There were the classical geometers, the cossist algebraists, the applied mathematicians and so on.

The geometers were fascinated by the humanistic idea and therefore praised any form of Greek science as superhuman while they did not seem to care much about the scientific work in other countries. One of the most outstanding representative of this group was Galileo, for whom ‘the book of the world was written in geometrical figures’.

The algebraists came mainly from Germany and Italy, and they very likely found the roots of their knowledge in the Arabic algebraic enterprise. They dealt mainly with arithmetical forms of problem solving and from there extended their interest from mere computation to formal algebra. Their methods and ambitions often interlinked with the applications of mathematics for the increasing commercial trade in Europe so that this branch grew steadily throughout the years. Yet this community was not a stable one since algebraists sometimes gained a part of their living by performing mathematics for the men of trade. Hence there was a constant pressure of publication of ideas and solutions of commonly known problems. For example the friendship between Cardano and Tartaglia was completely destroyed when Cardano published details of Tartaglia’s method of solving cubic equations, even though Cardano gave Tartaglia full credit for his work.

The applied mathematicians were mainly found in England, and they put their emphasis on all the ancient and newer works in all scientific disciplines which required mathematical background. They used their mathematical inventions for other scientific branches, and pure mathematics was not their goal. However, this did not hinder them from producing large amounts of excellent work on different fields of mathmatics. One of their best representatives was John Napier and his invention of the logarithm.

The last aspect worth mentioning to give a relatively complete picture of the mathematical world at that time is that there was no possibility for students to study any form of higher mathematics at universities. For most academical societies, there were only three higher sciences: law, medicine and theology. Only in a few German universities could students attend basic lectures about algebra and geometry. Therefore most of the mathematicians of that time gained their knowledge outside the university, largely on their own or with an individual correspondence with older mathematicians. This also led to a more individualized look in mathematics. Only after the establishment of the Savile Chair at Oxford in 1619 and the Lucasian Chair at Cambridge in 1664 did the status of mathematics start to change in the 17th century.
 
 
  

Fermat, Descartes, Mersenne, and the foundation of modern number theory in the 17th century

Here we enter the mathematical world of Fermat, where number theory became systematized and in a certain way rediscovered. With Fermat we say goodbye to the ancient and elementary number theory and arithmetic and welcome the modern number theory, which has lasted until today and was accompanied along the way by Fermat’s famous Last Theorem. Perhaps now,  after Wiles’ spectacular proof of Fermat’s Last Theorem, we can surmise a new era: the era of the postmodern number theory?