Elements of Geometric Sequence from One Divisible by Prime

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 $p$ be a prime number such that:

$p \divides a_n$

where $\divides$ denotes divisibility.

Then $p \divides a_1$.

In the words of Euclid:

If as many numbers as we please beginning from an unit be in continued proportion, by however many prime numbers the last is measured, the next to the unit will also be measured by the same.

(The Elements: Book $\text{IX}$: Proposition $12$)

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 $q \in \Z$.

Thus by hypothesis:

$p \divides q^n$

From Euclid's Lemma for Prime Divisors: General Result:

$p \divides q$

Hence the result.

$\blacksquare$

Historical Note

This proof is Proposition $12$ 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