It is widely stated that Fermat wrote his famous note on sums of powers ("Fermat's last theorem") in, or around, 1637. How do we know the date, if the note was only discovered after his death, in 1665?

My interest in this stems from the fact that if this is true, we can be absolutely certain that whatever proof Fermat in mind was wrong, and he must have noticed (or he would have mentioned it to his correspondents in later years). On the other hand, if the note had been written much later the reasoning would fail. I have used this argument previously in talks for the general public, rather acritically, and I would like to make sure it is sound.