Divisibility of Elements of Geometric Sequence from One where First Element is Prime

Theorem

Let $Q_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of length $n$ consisting of integers only.

Let $a_0 = 1$.

Let $a_1$ be a prime number.

Then the only divisors of $a_n$ are $a_j$ for $j \in \set {1, 2, \ldots, n}$.

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 prime, the greatest will not be measured by any except those which have a place among the proportional numbers.

(The Elements: Book $\text{IX}$: Proposition $13$)

Proof

From Form of Geometric Sequence of Integers from One, the elements of $Q_n$ are given by:

$Q_n = \tuple {1, a, a^2, \ldots, a^n}$

From Divisors of Power of Prime, each of $a_j$ for $j \in \set {1, 2, \ldots, n}$ are the only divisors of $a_n$.

$\blacksquare$

Historical Note

This proof is Proposition $13$ 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