20091110, 17:25  #1 
Jan 2005
Transdniestr
503 Posts 
Useful Identity?
I was fiddling around with WolframAlpha the other day and came across this identity:
(x^(n+3)+x^n + x + 1) = (x+1)*(x^(n+2)x^(n+1)+x^n+1) I hadn't seen it before and I was wondering if it's already known/in use for factoring larger numbers. This can be rearranged as (x^3+1)x^n+x+1 So, a more general identity could be derived from (x^(2m+1) + 1)x^n+ax+a 
20091110, 19:44  #2  
Nov 2003
2^{2}·5·373 Posts 
Quote:
It is trivial stuff from 1st/2nd year algebra. x^(n+3) + x^n = x^n(x^3+1). x^3+1 is trivially divisible by x+1. It is already known in the sense that it is so trivial that no mathematician would ever write it down or try to claim it as a "discovery". It is like the fact that 1+1=2. 

20091110, 20:44  #3 
Jan 2005
Transdniestr
767_{8} Posts 
Fair enough, it's trivial, too trivial to label it an identity even. I was just wondering if it could be useful like say the Aurifeuillian factorizations are.
Specifically, if it was useful in factoring a large number relatively near a perfect power. And, obviously, x^3 + 1 is a trivial bit of algebra. That's why I pointed out a general version of this IN THE NEXT LINE. 
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