# Quandry -- is the prime factorization of zero defined?

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

1. ### rpennerFully WiredValued Senior Member

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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

3. ### mathmanValued Senior Member

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1,831
Prime factors for 0 make no sense.