Book VIII · Proposition 3

VIII.3

If as many numbers as we please in continued proportion be the least of those which have the same ratio with them, the extremes of them are prime to one another.Heath, 1908

Every step, checked

• the ratio is a genuine onegiven
• a genuine progression is asked forgiven
• they are the least of their ratiogiven
• the extremes are prime to one anotherby VII.27
• and every measure of all of them is a unitby VII.21

What it needs, and what needs it

Needs: VII.21 VII.27

Depth: 3 steps of argument above the first principles. Parallel postulate: not needed.

What it takes on trust

Nothing. It draws no intersections and reads nothing off the picture.

The proposition as code

@proposition(
    "VIII.3",
    THEOREM,
    sample=progression,
)
def prop_VIII_3(p: int, q: int, count: int) -> Out:
    hypothesis("the ratio is a genuine one", p > 1 and q > 1, guard=True)
    hypothesis("a genuine progression is asked for", count >= 3, guard=True)
    terms = continued_proportion(1, (p, q), count)
    hypothesis("they are the least of their ratio", common_measure(terms) == 1)
    because(prop_VII_21, p, q) if coprime(p, q) else None
    because(prop_VII_27, p, q) if coprime(p, q) else None

    claim("the extremes are prime to one another", "VII.27",
          coprime(terms[0], terms[-1]))
    claim("and every measure of all of them is a unit", "VII.21",
          not any(all(measures(d, term) for term in terms)
                  for d in range(2, min(terms) + 1)))
    return Out()