[math-fun] sums of factors in GF(2)

[franktaw at netscape.net]

A. There are other sums to zero.  For example, (1+x+x^2) + (1+x^2+x^3) 
+ (1+x+x^4) + (1+x^3+x^4) = 0, and the parenthesized polynomials are 
all irreducible.  I doubt that these can be usefully characterized, but 
a few points can be noted:

1) Such a term can include x only if it also includes x+1.  The terms 
other than x all have 1 for their constant term, so there must be an 
even number of them.  Except for x+1, all have an odd number of 
non-zero coefficients, so if x was present and not x+1, the total 
number of non-zero coefficients would be odd, making the sum non-zero.

2) Likewise, if x+1 is present, x must be.  (Substitute x+1 for x in 
each polynomial.)

3) If x and x+1 are not present, the number of terms must be even; if 
they are present, it must be odd.  At least 4 terms are required.

B. Yes.  We can get every polynomial of degree <= 1: 0 from the empty 
set (sum of the prime divisors of 1), 1 from (x) + (1+x), and x and 
(1+x) are irreducible.  Since there is at least one irreducible 
polynomial of each degree n, which can be added to the polynomials of 
lower degree to get all polynomials of degree n, by induction every 
polynomial of degree n occurs.

Franklin T. Adams-Watters


-----Original Message-----
From: Marc LeBrun <mlb at well.com>

Recently I've been trying to exhort the Sequence Phanatiques to compute 
analogs of the sum-of-prime-factors (with and without multiplicity) in 
other arithmetics, such as Gaussian integers, GF(2), etc. 
 
I was wondering, specifically about GF(2), summing (ie XORing) the 
prime factors of N with multiplicity: 
 
Noting that only the square-free part of N matters, since the square 
parts sum to 0... 
 
A. Aside from the perfect squares (eg 5) are there any other N that sum 
to 0? Can they be characterized? 
 
B. If some sum S occurs for any N then it occurs infinitely, for all 
the square multiples of N. Does every value of S occur? 
 
 
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