Book IX · Proposition 15

IX.15

If three numbers in continued proportion be the least of those which have the same ratio with them, any two whatever added together will be prime to the remaining number.Heath, 1908

Every step, checked

• the ratio is a genuine onegiven
• the three are the least in their ratiogiven
• any two added together are prime to the remaining oneby VII.28

What it needs, and what needs it

Needs: VII.28

Depth: 1 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(
    "IX.15",
    THEOREM,
    sample=progression,
)
def prop_IX_15(p: int, q: int, count: int) -> Out:
    hypothesis("the ratio is a genuine one", p > 1 and q > 1, guard=True)
    terms = [q * q, p * q, p * p]
    hypothesis("the three are the least in their ratio", common_measure(terms) == 1)
    because(prop_VII_28, p, q) if coprime(p, q) else None

    claim("any two added together are prime to the remaining one", "VII.28",
          coprime(terms[0] + terms[1], terms[2])
          and coprime(terms[1] + terms[2], terms[0])
          and coprime(terms[0] + terms[2], terms[1]))
    return Out(terms=terms)