The Wikipedia Page on Fermat numbers states that $F_5$ was "fully factored" in 1732. This appears to be the same time that Euler found that any factor of a Fermat number $F_n$ was of the form
for some $k$ (later improved by Lucas) and found the smaller factor of $F_5$, the prime $641$, by trial and error. What isn't clear to me is whether Euler also verified that the larger factor $F_5 / 641 = 6,700,417$ was prime (it is).
C. Edward Sandifer's book How Euler did Even More contains the passage
Euler did not speculate in print on whether the other factor, 6,700,417, is prime. It is prime, but there is no evidence that Euler ever tried to find out.
but continues along the lines of how easy it is to prove, using the same tools Euler used to find the smaller factor.
You can't say a number is "fully factored" until you know all of its factors are prime.
I was originally going to ask whether Euler proved $6700417$ was prime, but in light of the above passage, I guess we'll never know.
Can we really say that $F_5$ was fully factored in 1732, or is there some date for the verification of $6700417$ that's more appropriate?
(user6530 pointed me to a cleaner version of the text than the Euler Archive preprint I'd originally linked to.)