I've been tinkering with Fermat Number factoring for several months.
Given my lack of math background and my preference to explore on my
own, I've been having fun rediscovering many things that are rather
simple (to most people here).
One thing I've been playing with is a simple transformation of the
F = (A*2^M+1)*(B*2^M+1)
F = A*B*2^(2*M)+A*2^M+B*2^M+1
(F-1)/2^M = (A*B*2^M) + (A+B)
A couple of days ago, I suddenly realized that if I could deduce what
value to subtract from (F-1)/2, I could simply factor the result to
determine the product of A*B. So I started to explore using small
values of A and B.
That's when I made the fatal error of transposing from
thinking "subtract (A+B)" and turned it to thinking "mod(A+B)". Given
small values of A and B, the product of A*B was small enough that
using mod(A+B) would extract the sum of A+B, leaving just the product
This excited me since I knew that K values typically are not prime
and consist of realitively small factors. I immediately setup a large
test using the unresolved portion of F12 to determine the
factorization of (F-1)/2^M mod(A+B). But it didn't take me too long
to realize that this wasn't giving me valid results.
At that point I went to bed and tried to ignore it long enough to get
some sleep. Then while drinking my morning coffee the next day, my
error suddenly tapped me on my shoulder. That's when I realized that
I was reducing the value by mod(A+B) instead of subtracting A+B. Talk
about a letdown!
So, I'm back to dabbling and tinkering with Fermat Number factoring.
My hind brain keeps telling me there is a way to reduce Fermat
Numbers. Given the long history of research on the subject and my
limited skills, I doubt there'll be any breakthrough. But you know
what? It sure is fun to try!
And no, I'm not certain why I'm sharing this. But it seemed amusing
to me and consequently I thought I'd share it for it's limited humor