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Something I learned a long time ago is this: if someone claims to have a proof, trying following the same proof in a different case where the conclusion is actually false, and see where their "proof" fails.

So let's try applying your reasoning to the example given in the linked article. There it shows that in the integers extended by sqrt(-5) we have 2x3=(1-sqrt(5))x(1+sqrt(5)). So using your specific reasoning:

    Let's take n=6

    f1 : 2 x 3
    f2 : (1-sqrt(5)) x (1+sqrt(5))
    p  : 2
    m  : 3
You now say:

    f2 does not contain p, ...

    ... and we cannot construct a prime p
        from other primes or composites.

    This means that f2 cannot be evenly
    divisible by p, as that would require
    it to have a prime factor of p which
    it does not.
But f2 is evenly divisible by p, despite not including p in the list of primes being multiplied together to give n. So your line of logic fails at this point.

This is actually assuming (something equivalent to) the FTA. The example shows a case where f2 is evenly divisible by p, so your deduction here is wrong.

It is subtle.

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