Primes

Quote:

Originally Posted by eyke

Show that if $m^4+4^n$ is prime, then $m$ is odd and $n$ is even, except when $m=n=1$ .

well, if $m$ is even, then obviously $4 \mid m^4 + 4^n$ and so $m^4 + 4^n$ cannot be prime. if $n=2k+1, \ k \geq 1,$ then putting $a=2^k$ we have $4^n=4^{2k+1}=4a^4.$

now $m^4+4^n=m^4+4a^4=(m^2+2a^2+2ma)(m^2+2a^2-2ma)$ and so $m^4+4^n$ cannot be prime because $m^2+2a^2-2ma=(m-a)^2+a^2 \geq a^2 > 1.$