If n>1, show that n^4 + 4^n is never a prime?
Okay, here we go. Prove that for all odd integers .
First, check the cases and separately.
For odd integers , I claim that . This can be proved by a slight variation of the method of induction.
When , .
Suppose for some odd integer .
Now consider . (NB: is the next odd integer.)
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Hence, for all odd integers , and it follows that .