From: peter.luschny@gmail.com   
      
   What are the prime factors of 0, really?   
      
   Many mathematicians have certainly thought about this question,   
   but what conclusion did they come to?   
      
   Among these were the developers of the remarkable CAS Maple.   
   For decades, they offered the 'numtheory' module and the   
   'factorset' function.   
      
   That then gave this answer:   
      
       restart; with(numtheory): seq(factorset(n), n = 0..9);   
      
       {0}, {}, {2}, {3}, {2}, {5}, {2, 3}, {7}, {2}, {3}   
      
   [1] The answer is: PrimeFactors(0) = {0}.   
      
   But one day at the daily meeting, someone said: "Yesterday   
   I read Apostol, and he says that every integer divides 0.   
   So factorset(0) should be the set of all primes, P, right?"   
   The developers were amazed but decided to write a new module.   
   This is now called 'NumberTheory' and the function 'PrimeFactors'.   
      
       restart; with(NumberTheory): seq(PrimeFactors(n), n = 1..9);   
      
       {}, {2}, {3}, {2}, {5}, {2, 3}, {7}, {2}, {3}   
       PrimeFactors(0); Error, cannot represent all prime factors of 0.   
      
   I suspect that the developers remembered that the set of prime   
   numbers, according to Euclid is infinite, and they wanted to   
   express this with the error message.   
      
   [2] The answer is: PrimeFactors(0) = P, but sorry.   
      
   But then there is a third answer: Consider the prime   
   factorization of n. In Maple, this function is called 'ifactors'   
   and looks like this:   
      
       seq(ifactors(n), n = 0..9);   
      
       [0, []], [1, []], [1, [[2, 1]]], [1, [[3, 1]]], [1, [[2, 2]]], ...   
      
   If you only look at the second coordinate of the respective element, you get:   
      
        seq({seq(f[1], f = ifactors(n)[2])}, n = 0..9);   
      
        {}, {}, {2}, {3}, {2}, {5}, {2, 3}, {7}, {2}, {3}   
      
   0 and 1 have no prime factors; all other integers have prime factors, as   
   indicated.   
      
   [3] The answer is: PrimeFactors(0) = {}.   
      
   Looks pretty convincing. Is there anything that speaks against it?   
      
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