It may work for odd numbers

Number Theory Level 2

True or False?

$\quad$ For integer $x>1$ , the expression $x^4 + 4^x$ cannot be a prime number .

False True

1 solution

Ossama Ismail
Apr 12, 2017

Answer is True

We will discuss two cases:-

$x$ is even $\implies x^4 + 4^x = even.$
x is odd

if $x$ is odd then we can write $x \ \text{as} \ 2n+1$ ; and the given equation can be written as

$(2n +1)^4 + 4^{(2n+1)} = (2n +1)^4 + 4.(2^n)^4$

Which is not a prime.