About Number Theory

    Number theory may very well be one of the oldest subjects of mathematics. Roughly, number theory is the mathematical treatment of questions related to the integers; that is, the numbers 0, -1, 1,-2, 2,-3, 3 ... and mankind has been manipulating them for thousands of years.

    The Ancient Greeks, in particular, Euclid (third century) and the Pythagoreans (sixth century) spent a considerable amount of attention to them; as well as Archimedes (third century). Indeed, one of the most important sets of numbers, the prime numbers; hold a key position in number theory since they are the building blocks of the integers; and perhaps the first question that comes to mind is whether there are infinitely many prime numbers. A proof of this amazing fact can be found in Euclid's famous book: The Elements.

        Pierre de Fermat and Leonard Euler rekindled interest in number theory in the seventeenth and eighteenth centuries by using new (among others, calculus-related) techniques to arrive at important new results. Each new theorem of course, has led to many new questions and conjectures; and one of the fascinating aspects of number theory is that many unresolved questions can be understood with only a minor background in the subject. Even today there are many open problems; and some have a substantial reward for a solution!

    After Fermat and Euler, Carl Friedrich Gauss, one of the greatest mathematicians of all time, gave the first modern treatment of number theory. He defined the notion of congruence, and distinguished its importance; in fact, it's his notation and approach of number theory that we use today. Gauss's many achievements in number theory are well documented; and it's he who coined the phrase "number theory is the queen of the sciences". Interestingly enough, even in an elementary course of number theory other fields of mathematics come into play, such as the complex numbers, geometry, and abstract algebra.

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