If you think about how modular exponentiation works (square and shift), then you will see why even k's are not needed. To test if some q=k*2^n+1 divides some Fm, you start with 2, and square mod q. If you get 1, then q divides F1. If not, you square again, and if 1, then q divides F2. If not, you square again, and if 1, then q divides F3. If not, you square again, and if 1, then q divides F4. If not, you square again, and if 1, then q divides F5.
Now you see why even k are not relevant?
Say I want to see if q=296*2^13+1 divides F11 (n has to be at least m+2 from a well known theorem). Repeating the above square+mod nine times, you get that this q divides F9 and you stop. Because two Fermat numbers can't share a factor. Exactly the same result you will get if you try to test if q=37*2^16+1 divides F14  you will run into "q divides F9" and stop after 9 steps (this is the same q).
Last fiddled with by LaurV on 20210105 at 09:30
