Book VII · Proposition 28

VII.28

If two numbers be prime to one another, the sum will also be prime to each of them; and, if the sum of two numbers be prime to any one of them, the original numbers will also be prime to one another.Heath, 1908

Every step, checked

• the numbers are prime to one anothergiven
• the sum is prime to each of themby VII.28
• and conversely, a sum prime to one makes the two prime to one anotherby VII.28

What it needs, and what needs it

Needs: nothing earlier.

Used by: IX.15

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.28",
    THEOREM,
    sample=coprime_pair,
)
def prop_VII_28(a: int, b: int) -> Out:
    hypothesis("the numbers are prime to one another", coprime(a, b))
    claim("the sum is prime to each of them", "VII.28",
          coprime(a + b, a) and coprime(a + b, b))
    claim("and conversely, a sum prime to one makes the two prime to one another",
          "VII.28", coprime(a + b, a) == coprime(a, b))
    return Out(total=a + b)