2^F4-1
 

I'm just wondering: since Mersenne included 2^257-1 in his list
of primes, and it is composite and completely factored,
what is the factorization status of 2^65537-1? The factoring
applet I tried doesn't try numbers that large. Have any small
factors been found? (I realize it must be composite.)

Yes indeed, although M65537 was proven composite by a Lucas-Lehmer test many years ago, it is only recently that two small factors have been found. See this thread:
[url]http://www.mersenneforum.org/showthread.php?t=8130[/url]
The smaller factor was apparently found in early November 2006 but had been missed earlier by a probably buggy version of Prime95. The cofactor is composite.

P-1 with B1 = 1e6, B2 = 1e9: no factor found.