Leonhard Euler

Leonhard Euler (1707-1783) was a remarkable scientist whose contributions have left their imprint on almost all branches of maths. His papers were re­warded ten times by prizes of the French Academy. His productivity was im­mense; it has been estimated that his collected works fill upward of 100 large volumes. One of his best known works Complete Introduction to Algebra (1770) contains much material on elementary number theory. Euler's factorization method applies only to numbers which in some way can be represented as a sum of two squares as, for instance, . It is possible to show that if a number can be represented as the sum of two squares, one can find all factorizations by Euler's method. Euler's method is capable of wide extensions. It leads to the theory of representations of numbers by means of a quadratic forms, i.e., .

Such representations can under certain conditions be used for factoring in the same manner as the special form .

It will carry us too far to discuss the great number of other aids and methods for factoring, some of them very ingenious. Considerable effort has been centred on the factorization of numbers of particular types. Some of them are numbers resulting from math problems of interest. Others have been selected because it is known for theoretical reasons that the factors must have a special form. Among the numbers that have been examinated in great detail one should mention the so-called binomial numbers where and are integers.

Georg Friedrich Bernhard Riemann

Although Euler had begun applying the methods of the calculus to number-theory problems, however, the German mathematician G. F. B. Riemann (1826-1866) is generally regarded as the real founder of analytic number theory. His personal life was modest and uneventful until his premature death from tuberculosis. According to the wish of his father he was originally destined to become a minister, but his shyness and lack of ability as a speaker made him abandon this plan in favour of math scholarship. At present he is recognized as one of the most penetrating and original math minds of the nineteenth century. In analytic number theory, as well as in many other fields of maths, his ideas still have a profound influence.

His starting point was a function now called Riemann's zeta function

This function he investigated in great detail and showed that its properties are closely connected with the prime-number distribution. On the basis of Riemann's ideas, the prime-number theorems were proved by other mathematicians. Much progress has been made in analytic number theory since that time, but it remains a peculiar fact that the key to some of the most essential problems lies in the so-called Riemann's hypothesis, the last of his conjectures about the zeta function, which has not been demonstrated. It states that the complex roots of the func­tion all have the real component .

Active Vocabulary

number	prime	sum
numeral	odd	difference	common	fraction
integer	even	product	decimal
fraction	factor	quotion	repeating
digit	divisor	ratio	non-repeating
series

whole	numbers	ring
counting		group
natural		field
rational		matrix
directed		numbers
complex		structures
imaginary		quaternions
transfinite		manifold
positive		isomorphism
negative

hierarchy of	structures
algebraic
topological		decimal	scale
order		duodecimal
mother		binary
multiple
isomorphic

measurable	existence	principle
immeasurable	uniqueness
commensurable	commutative
incommensurable	associative
	distributive
the least common multiple
the greatest common divisor