Elements of Geometric Sequence from One where First Element is not Power of Number

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Theorem

Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers.

Let $a_0 = 1$.

Let $k \in \Z_{> 1}$.

Let $a_1$ not be a power of $k$.


Then $a_m$ is not a power of $k$ except for:

$\forall m, k \in \set {1, 2, \ldots, n}: k \divides m$

where $\divides$ denotes divisibility.


In the words of Euclid:

If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be not square, neither will any other be square except the third from the unit and all those that leave out one. And, if the the number after the unit be not cube, neither will any other be cube except the fourth from the unit and all those that leave out two.

(The Elements: Book $\text{IX}$: Proposition $10$)


Proof

By Form of Geometric Sequence of Integers from One, the general term of $G_n$ can be expressed as:

$a_j = q^j$

for some $q \in \Z$.


Let $k \nmid m$.

Then by the Division Theorem there exists a unique $q \in \Z$ such that:

$m = k q + b$

for some $b$ such that $0 < b < k$.

Thus:

$a_m = a^{k q} a^b$

which is not a power of $k$.

$\blacksquare$


Historical Note

This proof is Proposition $10$ of Book $\text{IX}$ of Euclid's The Elements.


Sources