# Definition:Number Theory

## Definition

**Number theory** is the branch of mathematics which studies the properties of the natural numbers.

## Also defined as

Some sources allow that **number theory** studies the properties of all integers, not just the natural numbers, that is, the positive integers.

## Also known as

Older sources refer to **number theory** variously as:

**theory of numbers****higher arithmetic**or**the higher arithmetic****arithmetic**.

Carl Friedrich Gauss himself used the term **arithmetic**, after the manner of the ancient Greeks.

**Number theory** can also be seen described by the fanciful name **the Queen of Mathematics** for various whimsical reasons.

## Also see

- Results about
**number theory**can be found here.

## Historical Note

The field of number theory is considered by some to be one of the oldest branches of mathematics in history.

At the time of Pythagoras of Samos, there existed a mass of unstructured information on the subject dating back to the Babylonians and ancient Egyptians.

Pythagoras and his followers believed that all the phenomena in the Universe could be explained by the study of the natural numbers.

Some of the first serious results are found in Euclid's *The Elements*, thinly disguised as geometry.

Diophantus developed the ideas into a distinct branch of mathematics.

The field was founded in its modern form by Pierre de Fermat in his pioneering work in the $17$th century.

However, most of his discoveries are known about only because he wrote about them to his friends, or (famously) jotted them down in the margins of his copy of Diophantus's *Arithmetica*.

For many of these, his proofs were never recorded, and when he died they were lost forever.

Nobody else was able to follow him until Euler and Lagrange in the following century.

The field was properly placed on a firm footing by the work of Carl Friedrich Gauss in his *Disquisitiones Arithmeticae*.

The field was advanced significantly by Augustin Louis Cauchy.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs - 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**arithmetic**:**2.**(**the higher arithmetic**) - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**higher arithmetic** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.16$: The Sequence of Primes - 1994: H.E. Rose:
*A Course in Number Theory*(2nd ed.) ... (next): Preface to first edition - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $7$: Patterns in Numbers: Number Theory - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**number theory (higher arithmetic)**