Number Theory
What can be more basic in mathematics than integer numbers like 1, 2, 3, and so on? Thus, one should hardly be surprised that the study of such numbers goes back to the very beginnings of formal mathematics, to Greek mathematicians of the sixth century B.C. Today, this field of mathematics is known as number theory. Number theory involves the analysis of the properties of the natural numbers (1, 2, 3, etc.) and, more generally, of all integers, positive or negative, and zero (...-2,-1, 0, 1, 2...).
The Greek natural philosopher Pythagoras of Samos carried out some of the earliest and most primitive research on number theory. Indeed, Pythagoras became virtually obsessed by the natural numbers and taught that they formed the basis of the natural universe. Although many of Pythagoras' ideas were more mystical and rational, his interest in numbers led to a great deal of early study in the field. For example, early Pythagoreans were the first to discover that 2 can not be expressed as the ratio of two whole numbers.
The properties of numbers also interested the third-century B.C. Greek mathematician Euclid. He was especially interested in the prime numbers and was the first to show that the number of primes is infinite. Still, Euclid's fame rests not so much on his own work as on his monumental accomplishment of bringing together all known mathematical knowledge in a single textbook, his Elements. He devoted books seven to nine of Elements to number theory.
An important contribution to number theory is found in the work of yet a third Greek mathematician, Diophantus. Diophantus looked for a method of finding integer solutions to indeterminate equations, equations that lack sufficient information to produce a single discrete set of answers. The equation x + y = 5 is such an equation. Diophantus discovered that many indeterminate equations can be reduced to a form where a certain category of answers is known even though a specific answer is not.
As with many other forms of knowledge, progress in number theory languished during the Dark Ages. It was not until the seventeenth century that mathematicians once again turned their attention to number theory in earnest. Some of the most important developments at that time were found in the work of the Frenchman, Pierre de Fermat. Fermat attacked a number of problems with which the Greeks had worked and extended their analysis in a number of directions. He developed many new theorems and concepts in number theory. One of the problems for which he is most famous involves the indeterminant equation xn + y n = zn. The equation obviously has many integer solutions for n = 1 and the Pythagoreans had thoroughly studied the case of n = 2. But what was the situation for n > 2? Fermat claimed that the equation has no solution for n = 3, n = 4, and, as stated in his famous "last theorem," any value of n > 2. Fermat's proof, however, was never found and his claim was to become one of the most enduring challenges in all of mathematics.
Fermat's work provided such a solid framework for modern number theory that he is sometimes called the founder of this field of mathematics. Still, his work remained largely without influence until re-examined by the Swiss mathematician Leonhard Euler in the mid-eighteenth century. Euler attacked a number of problems originally raised by Fermat and found solutions for them. For example, he was able to prove the impossibility of finding integer solutions for the equation xn + yn = zn for n = 3. Later, Sophie Germain proved a special case of Fermat's equation for all prime exponents less than 100, and the French mathematician Adrien-Marie Legendre proved the general case for n = 5. Mathematicians continued to search for a proof or disproof of the final theorem with only limited success. Using computers, confirmation for values up to n = 235,747,889 under various conditions were obtained, but no universal proof had been discovered until 1993, when Andrew Wiles, a mathematician at Princeton University, announced to a stunned audience of mathematics professors that he had solved Fermat's last theorem. Wiles finally published this proof in 1995, after having to work an extra two years to patch an error in the original version.
If Fermat and Euler laid the foundations for modern number theory, the chief architect of its final structure was the German mathematician Carl Friedrich Gauss. Regarded as one of the greatest mathematicians of the nineteenth centuries, Gauss developed most of the methods and many of the theorems used in number theory today. He published many of these results in Latin in his monumental book Disquisitiones Arithmeticae in 1801, in which he applied the methods of algebra, calculus, and function theory to problems of number theory. Two of his most important accomplishments involved the calculation of the distribution of prime numbers among all the integers and the way in which complex numbers can be factored. Gauss also made fundamental contributions to the theory of congruences and to the study of algebraic numbers.
One of the most prolific researchers in number theory in the twentieth century was the Hungarian mathematician Paul Erdös (1913-1996). In 1949 he published one of the first elementary proofs of the prime number theorem, and at his death had written more than 1,500 mathematical papers in number theory, combinatorics, and discrete mathematics. The variety and complexity of number theory problems continue to challenge mathematicians. There appears to be no limit to the variety of questions that one can ask and the range of relationships one can find among the integers.
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