Book IX · Proposition 31

IX.31

If an odd number be prime to any number, it will also be prime to the double of it.Heath, 1908

Every step, checked

• the number is oddgiven
• it is prime to the othergiven
• it is prime to the double of it alsoby IX.30

What it needs, and what needs it

Needs: IX.30

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.31", THEOREM,
             sample=lambda rng: (2 * rng.randint(1, 30) + 1, rng.randint(2, 40)))
def prop_IX_31(odd: int, other: int) -> Out:
    hypothesis("the number is odd", not measures(2, odd))
    hypothesis("it is prime to the other", coprime(odd, other))
    because(prop_IX_30, odd, other)

    claim("it is prime to the double of it also", "IX.30", coprime(odd, 2 * other))
    return Out()