It's a bad notation to define a number to be "positive" if it lies within some sector, for then products of positive numbers need not be positive. The only situation I can think of where it's sensible to adopt a convention for the choice of associate is in a computer algebra factorization function. Then a number will always factor the same way into primes and a unit, assuming of course that we are working in a UFD (unique factorization domain). -- Gene
________________________________ From: Marc LeBrun <mlb@well.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Thursday, June 9, 2011 3:01 PM Subject: Re: [math-fun] Some Sum
="Marc LeBrun" <mlb@well.com> This is interesting, but I'm still having trouble with "THE first quadrant".
Risking belaboring the obvious in the pursuit of clarity:
= Kieran Smallbone For example
2+4i = (-i).((1+i)^2).(1+2i)
where the primes 1+i and 1+2i are "positive" because both they lie
in the top right quadrant.
But x=1-i has no "positive" Gaussian integer divisors at all.
Are we then to take the sum of the divisors of x to be zero?
Similarly, is the sum of the divisors of 2 then to be 1+(1+i)+2 = 4+i?
(Heh. Then there are no even perfect naturals. Are there any at all?)