Four perfect fifths (minus two octaves) doesn't equal a major third

If we start with a pitch of frequency F, the frequency of a pitch a just perfect fifth higher is F * 3/2.
Another fifth up takes us to F * 3/2 * 3/2, or F * 9/4.
Next is F * 27/8, and one more is F * 81/16.
A just major seventeenth (a just major third plus two octaves) is F * 5, or F * 80/16, which is a bit flatter.