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# 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*

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