# Sum of Pair of Elements of Geometric Sequence with Three Elements in Lowest Terms is Coprime to other Element

## Theorem

Let $P = \tuple {a, b, c}$ be a geometric sequence of integers in its lowest terms.

Then $\paren {a + b}$, $\paren {b + c}$ and $\paren {a + c}$ are all coprime to each of $a$, $b$ and $c$.

In the words of Euclid:

If three numbers in continued proportion be the least of those which have the same ratio with them, any two whatsoever added together will be prime to the remaining number.

## Proof

Let the common ratio of $P$ in canonical form be $\dfrac q p$.

$P = \tuple {p^2, p q, q^2}$

Then:

 $\ds p$ $\perp$ $\ds q$ Definition of Canonical Form of Rational Number $\ds \leadsto \ \$ $\ds q$ $\perp$ $\ds \paren {p + q}$ Numbers are Coprime iff Sum is Coprime to Both $\ds \leadsto \ \$ $\ds q$ $\perp$ $\ds p \paren {p + q}$ Integer Coprime to all Factors is Coprime to Whole $\ds \leadsto \ \$ $\ds q^2$ $\perp$ $\ds p \paren {p + q}$ Square of Coprime Number is Coprime $\ds \leadsto \ \$ $\ds q^2$ $\perp$ $\ds p^2 + p q$ factorising $\ds \leadsto \ \$ $\ds c$ $\perp$ $\ds a + b$

Similarly:

 $\ds \leadsto \ \$ $\ds p^2$ $\perp$ $\ds p q + q^2$ $\ds \leadsto \ \$ $\ds a$ $\perp$ $\ds b + c$ factorising

Then:

 $\ds p + q$ $\perp$ $\ds p$ Numbers are Coprime iff Sum is Coprime to Both $\, \ds \land \,$ $\ds p + q$ $\perp$ $\ds q$ Numbers are Coprime iff Sum is Coprime to Both $\ds \leadsto \ \$ $\ds \paren {p + q}^2$ $\perp$ $\ds p q$ as above $\ds \leadsto \ \$ $\ds p^2 + q^2 + 2 p q$ $\perp$ $\ds p q$ factorising $\ds \leadsto \ \$ $\ds p^2 + q^2$ $\perp$ $\ds p q$ $\ds \leadsto \ \$ $\ds b$ $\perp$ $\ds a + c$

$\blacksquare$

## Historical Note

This proof is Proposition $15$ of Book $\text{IX}$ of Euclid's The Elements.