Quandry -- is the prime factorization of zero defined?

Discussion in 'Physics & Math' started by rpenner, Mar 8, 2012.

  1. rpenner Fully Wired Valued Senior Member

    Messages:
    4,833
    What is the factorization of zero? Is there a good representation of it as a product of powers of primes or should one treat it as a special case?

    \( 0 = \quad \lim_{n\to\infty} \prod_{p \in \mathbb{P} \, \wedge \, p < n } p^{-1} \; = \quad \quad \lim_{n,m\to\infty} \prod_{p \in \mathbb{P} \, \wedge \, p < n } p^{-m} \)

    So should the "prime factorization" of 0 be:
    a) undefined
    b) \(0^{\tiny 1}\), with special rules for manipulating the representation when multiplied by another number
    c) \(2^{\tiny -1} \, \times \, 3^{\tiny -1} \, \times \, 5^{\tiny -1} \, \times \, 7^{\tiny -1} \, \times \, 11^{\tiny -1} \, \times \, \dots\), with special rules for manipulating the representation when multiplied by another number
    d) \(2^{\tiny -\infty} \, \times \, 3^{\tiny -\infty} \, \times \, 5^{\tiny -\infty} \, \times \, 7^{\tiny -\infty} \, \times \, 11^{\tiny -\infty} \, \times \, \dots\)

    Is the number of the distinct prime factors of 0:
    a) undefined
    b) 0
    c) 1
    d) infinite

    Is the sum of the distinct prime factors of 0:
    a) undefined
    b) 0
    c) infinite

    Are there other answers that make sense?

    Mathematica's FactorInteger[0] gives {{0,1}} which corresponds to \(0^{\tiny 1}\), but PrimeQ[0] is false and PrimeNu[0] is undefined.

    I'm beginning to think that in a general reference, "undefined" is the best answer to all of these, so I should withdraw results that relied on Total[Map[First,FactorInteger[n]]] when n = 0
     
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  3. mathman Valued Senior Member

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    Prime factors for 0 make no sense.
     
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  5. Absane Rocket Surgeon Valued Senior Member

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    8,989
    I think it's a pointless exercise to try to investigate the factorizing of 0. ALL complex numbers divide 0, even 0.
     
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