More generally, let S be any set of Gaussian primes such that the set of all Gaussian primes is the disjoint union of S, -S, iS, -iS. Then we may call S the "standard primes", and any Gaussian integer can be uniquely factored into standard primes and a unit. There doesn't seem to be any natural choice of S, and "top right quadrant" or "Conway-Guy" are as good as any other convention. The sum of divisors function, which evaluates to a single number, because it depends on the choice of S, seems to me to be an uninteresting object. If instead the function evaluates to the finite set of sums of all possible choices of associates, then it would not involve a choice of S. Perhaps, in this form, a sum of divisors set function would have interesting mathematical properties. -- Gene
________________________________ From: Hans Havermann <pxp@rogers.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Thursday, June 9, 2011 9:30 PM Subject: Re: [math-fun] Some Sum
Marc LeBrun:
you can't fully factorize 5 into "top-right-quadrant" primes exclusively
Mathematica factors Gaussian integers and I just plotted all factors of x + y i for integers x and y from -100 to +100. With the exception of -1 and -i, all other factors are in the top-right quadrant. More specifically:
In[1]:= FactorInteger[5,GaussianIntegers->True] Out[1]= {{-I,1},{1+2 I,1},{2+I,1}}
I'm going to guess that the "positive" Gaussian quadrant illustrated by Conway and Guy can be rotated into the *top-right* quadrant without loss of the unique-factorization principle.
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