Fermat Numbers have the form Fk=22k+1
Mersenne Numbers have the form Mk=2k-1
*NB: these are not the sets of Fermat Primes and Mersenne Primes.
I observe the following relation: the product of the sequence of the first n Fermat Numbers is Mersenne Number 2n
PRODUCT(i=0..n-1) Fi = M2n
Is this a well-known identity? I'd rather not try to prove this if it is.
I verified it in a WxMaxima loop by multiplying the Fi and factoring the (product+1),
until Lisp got a value stack overflow, and of course it all checked out, or I wouldn't be posting this.
The last result produced before the overflow was n=18, which is M262144,
a number 78'914 digits long (ending in 5 as it should, since all the Fi are odd and F1 = 5).
EDIT: I saw no mention of this identity in OEIS.