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

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