Definition:Prime Decomposition
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Definition
Let $n > 1 \in \Z$.
From the Fundamental Theorem of Arithmetic, $n$ has a unique factorization of the form:
- $n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$
where $p_1 < p_2 < \cdots < p_r$ are distinct primes and $k_1, k_2, \ldots, k_r$ are positive integers.
This unique expression is known as the prime decomposition or prime factorization of $n$.
Linguistic Note
The UK English spelling of prime factorization is prime factorisation.
Sources
- George E. Andrews: Number Theory (1971): $\S 2.4$: Exercise $3$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 24$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 13$
