## You are here

Homefundamental theorem of arithmetic

## Primary tabs

# fundamental theorem of arithmetic

Each positive integer $n$ has a unique decomposition as a product

$n=\prod_{{i=0}}^{l}{p_{i}}^{{a_{i}}}$ |

of positive powers of its distinct positive prime divisors $p_{i}$. The prime divisor of $n$ means a (rational) prime number dividing $n$. A synonymous name is prime factor.

The decomposition is unique up to the order of the prime divisors and for $n=1$ is an empty product.

For some results it is useful to assume that $p_{i}<p_{j}$ whenever $i<j$.

The FTA was the last of the fundamental theorems proven by C.F. Gauss. Gauss wrote his proof in “Discussions on Arithmetic” (Disquisitiones Arithmeticae) in 1801 formalizing congruences. Euclid and Greeks used prime properties of the FTA without rigorously proving its existence. It appears that the fundamentals of the FTA were used centuries before, and after, the Greeks within Egyptian fraction arithmetic. Fibonacci, for example, wrote in Egyptian fraction arithmetic, used three notations to detail Euclidean and medieval factoring methods.

## Mathematics Subject Classification

11A05*no label found*17B66

*no label found*17B45

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias