true for n=1,2,3,4,5
true for n=1,2
true for n=1,2,3,4
always true

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These numbers are called Fermat Numbers , and for a long time they were thought to be automatically prime. Leonard Euler found that the $5^{\text{th}}$ Fermat Number, $4294967297$ , was not in fact prime. being divisible by $641$ . Thus the conjecture that all numbers of this form are prime is false, and it is only valid for $n={0,1,2,3,4}$