Example: Fermat Numbers

The zeroth through 4th Fermat numbers are indeed prime. Unfortunately, the 5th Fermat number:
   4,294,967,297 = 641 × 6,700,417
Only the zeroth through 4th numbers are known to be prime.

Many are known to be composite; the largest known is the 23,471st.

The smallest Fermat number whose primality is not known is the 24th.