Book VII · Proposition 29

VII.29

Any prime number is prime to any number which it does not measure.Heath, 1908

Every step, checked

• the first is primegiven
• it does not measure the secondgiven
• a prime is prime to any number it does not measureby VII.29

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(
    "VII.29",
    THEOREM,
    sample=lambda rng: (rng.choice([p for p in range(2, 60) if is_prime(p)]),
                        rng.randint(2, 200)),
)
def prop_VII_29(p: int, n: int) -> Out:
    hypothesis("the first is prime", is_prime(p))
    hypothesis("it does not measure the second", not measures(p, n))
    claim("a prime is prime to any number it does not measure", "VII.29",
          coprime(p, n))
    return Out()