# Fundamental Theorem of Arithmetic

## Theorem

For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquely up to the order in which they appear.

## Proof

In Integer is Expressible as Product of Primes it is proved that every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes.

In Prime Decomposition of Integer is Unique, it is proved that this prime decomposition is unique up to the order of the factors.

$\blacksquare$

## Also known as

This result is otherwise known as the **unique factorization theorem**.

However, this is the name of the equivalent result for the general Euclidean domain and it can be argued that the names are best kept separate.

## Historical Note

The Fundamental Theorem of Arithmetic was first proved by Carl Friedrich Gauss in his *Disquisitiones Arithmeticae* in $1801$.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs - 1958: Martin Davis:
*Computability and Unsolvability*... (previous) ... (next): Appendix $1$: Some Results from the Elementary Theory of Numbers: Theorem $10$ - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $3$: The Integers: $\S 13$. Unique Factorisation: Theorem $22$ - 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {2-4}$ The Fundamental Theorem of Arithmetic: Theorem $\text {2-5}$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Properties of the Natural Numbers: $\S 24$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 13$: The fundamental theorem of arithmetic - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 2.2$: Divisibility and factorization in $\mathbf Z$: Theorem $4$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $1$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.16$: The Sequence of Primes: Theorem $1$ - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $21$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1$