I have two expressions (both of which have a term raised to the power of $n$) and I am trying to prove that they can't be prime numbers at the same time for $n>2$.
I can't post the expressions, but I was wondering if there was someway to prove it by saying: "Let the first and the second be prime numbers then because their product is equal to something they can't be prime numbers at the same time.
Is their a statement or law that I can use?
On
Of any three consecutive integers, one is divisible by $3$. In particular, one of $2^n-1$, $2^n$, and $2^n+1$ is divisible by $3$. But $2^n$ is not divisible by $3$, so one of $2^n-1$ and $2^n+1$ is divisible by $3$.
If $n\gt 2$, then $2^n-1$ and $2^n+1$ are both bigger than $3$. One of them is divisible by $3$ and greater than $3$, so is not prime.
Hint: Primes other than 2,3 always have the form $6k + 1$ or $6k + 5$ for $k \in \mathbb Z$.
To prove this, notice that numbers that have the form $6k, 6k+2$ or $6k+4$ are divisible by 2, and numbers that have the form $6k+3$ are divisible by 3.