# Definition:Number Theory/Historical Note

## Historical Note on Number Theory

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

- 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.3$ Early Number Theory - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($1601$ – $1665$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.25$: Gauss ($1777$ – $1855$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.26$: Cauchy ($1789$ – $1857$) - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $7$: Patterns in Numbers: Number Theory