Decomposition into Even-Odd Integers is not always Unique
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Theorem
For every even integer $n$ such that $n > 1$, if $n$ can be expressed as the product of one or more even-times odd integers, it is not necessarily the case that this product is unique.
Proof
Let $n \in \Z$ be of the form $2^2 p q$ where $p$ and $q$ are odd primes.
Then:
- $n = \paren {2 p} \times \paren {2 q} = 2 \times \paren {2 p q}$
A specific example that can be cited is $n = 60$:
- $60 = 6 \times 10$
and:
- $60 = 2 \times 30$.
Each of $2, 6, 10, 30$ are even-times odd integers:
\(\ds 2\) | \(=\) | \(\ds 2 \times 1\) | ||||||||||||
\(\ds 6\) | \(=\) | \(\ds 2 \times 3\) | ||||||||||||
\(\ds 10\) | \(=\) | \(\ds 2 \times 5\) | ||||||||||||
\(\ds 30\) | \(=\) | \(\ds 2 \times 15\) |
Every $n \in \Z$ which has a divisor in that same form $2^2 p q$ can similarly be decomposed non-uniquely into even-times odd integers.
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-4}$ The Fundamental Theorem of Arithmetic: Exercise $1$