Elements of Geometric Sequence from One which are Powers of Number
Jump to navigation
Jump to search
Theorem
Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers.
Let $a_0 = 1$.
Then:
- $\forall m, k \in \set {1, \ldots, n}: k \divides m \implies a_m$ is a power of $k$
where $\divides$ denotes divisibility.
In the words of Euclid:
- If as many numbers as we please beginning from an unit be in continued proportion, the third from the unit will be square, as will those which successively leave out one; the fourth will be cube, as will also all those which leave out two; and the seventh will be at once cube and square, as will also those which leave out five.
(The Elements: Book $\text{IX}$: Proposition $8$)
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 $k, m \in \set {1, 2, \ldots, n}$ such that $k \divides m$.
By definition of divisibility:
- $\exists r \in \Z: m = r k$
Then:
\(\ds a_m\) | \(=\) | \(\ds q^m\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds q^{r k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {q^r}^k\) |
That is, $a_m$ is a power of $k$.
$\blacksquare$
Historical Note
This proof is Proposition $8$ of Book $\text{IX}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{IX}$. Propositions