Expression for Integers as Powers of Same Primes/General Result

Theorem
Let $a_1, a_2, \dotsc, a_n \in \Z$ be integers.

Let their prime decompositions be given by:


 * $\displaystyle a_i = \prod_{\substack {p_{i j} \mathop \divides a_i \\ \text {$p_{i j}$ is prime} } } {p_{i j} }^{e_{i j} }$

Then there exists a set $T$ of prime numbers:
 * $T = \set {t_1, t_2, \dotsc, t_v}$

such that:
 * $t_1 < t_2 < \dotsb < t_v$


 * $\displaystyle a_i = \prod_{j \mathop = 1}^v {t_j}^{g_{i j} }$

Proof
The proof proceeds by induction.

For all $n \in \Z_{\ge 2}$, let $\map P n$ be the proposition:
 * for all $a_i \in \set {a_1, a_2, \ldots, a_n}$: there exists a set $T = \set {t_1, t_2, \dotsc, t_v}$ of prime numbers such that $t_1 < t_2 < \dotsb < t_v$ such that:
 * $\displaystyle a_i = \prod_{j \mathop = 1}^v {t_j}^{g_{i j} }$

Basis for the Induction
$\map P 2$ is the case:

there exist prime numbers $t_1 < t_2 < \dotsb < t_v$ such that:

This has been proved in Expression for Integers as Powers of Same Primes.

Thus $\map P 2$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:
 * for all $a_i \in \set {a_1, a_2, \ldots, a_k}$: there exists a set $T = \set {t_1, t_2, \dotsc, t_v}$ of prime numbers such that $t_1 < t_2 < \dotsb < t_v$ such that:
 * $\displaystyle a_i = \prod_{j \mathop = 1}^v {t_j}^{g_{i j} }$

from which it is to be shown that:
 * for all $a_i \in \set {a_1, a_2, \ldots, a_{k + 1} }$: there exists a set $T' = \set {t_1, t_2, \dotsc, t_w}$ of prime numbers such that $t_1 < t_2 < \dotsb < t_w$ such that:
 * $\displaystyle a_i = \prod_{j \mathop = 1}^w {t_j}^{g_{i j} }$

Induction Step
This is the induction step:

Let $E = \set {q_i: q_i \divides a_{k + 1}, \text {$q_i$ is prime} }$.

Then let:
 * $T' = E \cup T$

and let the elements of $T$ be renamed as:
 * $T' = \set {t_1, t_2, \ldots, t_w}$

where all the $t_1, t_2, \dotsc, t_w$ are distinct, and:
 * $t_1 < t_2 < \dotsb < t_w$

Then we have that:

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore, for all $n \in \Z_{\ge 2}$:
 * for all $a_i \in \set {a_1, a_2, \ldots, a_n}$: there exists a set $T = \set {t_1, t_2, \dotsc, t_v}$ of prime numbers such that $t_1 < t_2 < \dotsb < t_v$ such that:
 * $\displaystyle a_i = \prod_{j \mathop = 1}^v {t_j}^{g_{i j} }$