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

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## 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.*

(*The Elements*: Book $\text{IX}$: Proposition $15$)

## Proof

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

By Form of Geometric Sequence of Integers in Lowest Terms:

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

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*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{IX}$. Propositions