Book IX · Proposition 6

IX.6

If a number by multiplying itself make a cube number, it will itself also be cube.Heath, 1908

Every step, checked

• the number is genuinegiven
• its square is cubegiven
• the number itself is cubeby IX.6
• and no number whose square is cube fails to be oneby IX.6

What it needs, and what needs it

Needs: nothing earlier.

Depth: 0 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.6",
    THEOREM,
    sample=lambda rng: (rng.randint(2, 8) ** 3,),
)
def prop_IX_6(a: int) -> Out:
    hypothesis("the number is genuine", a > 1, guard=True)
    hypothesis("its square is cube", is_cube(a * a))
    claim("the number itself is cube", "IX.6", is_cube(a))
    claim("and no number whose square is cube fails to be one", "IX.6",
          all(is_cube(n) for n in range(2, 300) if is_cube(n * n)))
    return Out()