Fermat Prime Conjecture

False Conjecture
All numbers of the form $2^{\paren {2^n} } + 1$, where $n = 0, 1, 2, \ldots$ are prime.

This was postulated by.

Refutation
Although true for $n = 0, 1, 2, 3, 4$, the conjecture fails for $n = 5$.

From Prime Decomposition of $5$th Fermat Number:
 * $2^{\paren {2^5} } + 1 = 641 \times 6 \, 700 \, 417$

Also see

 * Definition:Fermat Number
 * Definition:Fermat Prime


 * Prime Decomposition of $5$th Fermat Number
 * Prime Decomposition of $6$th Fermat Number
 * Prime Decomposition of $7$th Fermat Number
 * Prime Decomposition of $8$th Fermat Number
 * Prime Decomposition of $9$th Fermat Number