It's part of a larger problem but this is what I got left;
if 3^a - 2^a is a prime number, then a is a prime number.
If anyone could point me to the right direction to prove/disprove this, it would be nice. Thanks!
I do not know if i am going to help you, but:
Why dont you try to expand the identity as follow:
From binomial expansion we have:
And then we have the equality:
From there i have nothing more...
Some ideas:
If we rise both sides to a expand the right side and the again rise to a,
mayby we could compare the addition terms.
or
if we try to simplify left side by divisions of 2.
again i do not know if this helps to prove (or disprove) your identity.
The contrapositive is easier to prove.
So you need to show that:
If is NOT a prime number, then is NOT a prime number.
Recall that a prime number is one that only has 1 and itself as factors.
So to show that is not prime, we would have to show that it has a common factor.
First, if is not prime, then it can be expressed as , where .
So
Therefore
.
Now we make use of the factorisation rule
.
So
.
Thus is not prime.
So, if is not prime, then is not prime.
Therefore, if IS prime, then is prime.
I got to that and the factors when I tried the problem a few hours later but I wasn't sure if I was right. Can you just say that ? I mean, (or doesnt it count because it CAN be written and then **** it up?) would'nt be a prime and it wouldn't screw up the factor later on. I dont need to show that the factor with the sum is prime if for example and or any other number where and prime?
Sorry for the stupid questions, I'm kinda new to this proving thing. Really appreciate the help!