What Is Number Theory?

Tracey_Parece | Jun 23, 2010 |

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Number Theory

There are several different fields of number theory. Once called "arithmetic" or "higher arithmetic," number theory involves the properties of integers, rational numbers, irrational numbers, and real numbers. Imaginary and complex numbers are sometimes included in number theory. Different fields of number theory include the following: algebraic number theory, analytic number theory, arithmetic algebraic geometry, arithmetic dynamics, combinatorial number theory, computational number theory, elementary number theory, geometry of numbers, and modular forms.

Number theory was studied in the 3rd century with the Greek Mathematician Diophantus (although Egyptian and Babylonian records of number theory predate those of the Greeks). Indian, Islamic, and Early European mathematicians continued the study of number theory. The origins of early European number theory are found in 16^th and 17^th century Europe. One mathematician who studied number theory in the 1600s was Pierre de Fermat. He expounded upon theories founded centuries earlier by Greek mathematician Diophantus:

"He (Pierre de Fermat) gave a proof of the statement made by Diophantus that the sum of the squares of two integers cannot be of the form 4n - 1; and he added a corollary which I take to mean that it is impossible that the product of a square and a prime of the form 4n - 1 [even if multiplied by a number prime to the latter], can be either a square or the sum of two squares. For example, 44 is a multiple of 11 (which is of the form 4 × 3 - 1) by 4, hence it cannot be expressed as the sum of two squares. He also stated that a number of the form a² + b², where a is prime to b, cannot be divided by a prime of the form 4n - 1 (Ball, W.W. Rouse)."

Modern number theory began at the turn of the 19^th century with mathematicians Adrian Marie Legendre and Karl Friedrich Gauss (sometimes referred to as the prince of mathematics). Legendre wrote about the subject of number theory in Recherches d'analyse indéterminée in 1785. Gauss wrote Disquisitiones Arithmeticae in 1801 to systemize the study of number theory when he was only 24 years old.

Number theory continues to be studied and researched today. Ulam's Spiral (see Fig. 1.1) was not created until 1963 when Stanislaw Ulam discovered that circling all the prime numbers on a chart of integers (listing the numbers in a spiral with 1 at the center) would form a pattern. The prime numbers on such an illustration lie on diagonal lines. Fermat's Last Theorem was not proven until 1994 by a mathematician named Andrew Wiles. There remain unsolved problems in number theory so it is a field that will no doubt continue to be studied and researched into infinity.