First Element of Geometric Sequence not dividing Second/Proof 1
Theorem
Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers of length $n$.
Let $a_0$ not be a divisor of $a_1$.
Then:
- $\forall j, k \in \set {0, 1, \ldots, n}, j \ne k: a_j \nmid a_k$
That is, if the initial term of $P$ does not divide the second, no term of $P$ divides any other term of $P$.
In the words of Euclid:
- If there be as many numbers as we please in continued proportion, and the first do not measure the second, neither will any other measure any other.
(The Elements: Book $\text{VIII}$: Proposition $6$)
Proof
Let $P_a = \tuple {a_0, a_1, \ldots, a_n}$ be a geometric sequence of natural numbers such that $a_0 \nmid a_1$.
Aiming for a contradiction, suppose $a_0 \divides a_k$ for some $k: 2 \le k \le n$.
Let $b_0, b_1, \ldots, b_k$ be the least natural numbers which have the same common ratio as $a_0, a_1, \ldots, a_k$.
These can be found by means of Proposition $33$ of Book $\text{VII} $: Least Ratio of Numbers.
From Proposition $14$ of Book $\text{VII} $: Proportion of Numbers is Transitive
- $a_0 : a_k = b_0 : b_k$
Also:
- $a_0 : a_1 = b_0 : b_1$
and so as $a_0 \nmid a_1$ it follows by Book $\text{VII}$ Definition $20$: Proportional:
- $b_0 \nmid b_1$
From One Divides all Integers it follows that:
- $b_0 \ne 1$
From Proposition $3$ of Book $\text{VIII} $: Construction of Sequence of Numbers with Given Ratios:
- $b_0 \perp b_k$
But as:
- $a_0 : a_k = b_0 : b_k$
it follows that:
- $a_0 \nmid a_k$
Now suppose $a_j \divides a_k$ such that $0 < j < k$.
Let $b_j, \ldots, b_k$ be the least natural numbers which have the same common ratio as $a_j, \ldots, a_k$.
These can be found by means of Proposition $33$ of Book $\text{VII} $: Least Ratio of Numbers.
From Proposition $14$ of Book $\text{VII} $: Proportion of Numbers is Transitive:
- $a_j : a_k = b_j : b_k$
The other cases follow similarly.
$\blacksquare$
Historical Note
This proof is Proposition $6$ of Book $\text{VIII}$ 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 $\text{VIII}$. Propositions