Find all positive integers n for which 3n−4, 4n−5, and 5n−3 are all prime numbers.
We’re asked to find all integers satisfying are all primes.
The hidden trick behind this problem is that is even. Now, suppose that the sum of three prime integers turned out to be even, let’s enumerate all the ways to add three numbers to find out how this is possible.
So either exactly one of is even or that all of are even.
Let’s look at the case in which each of the primes is even. The only even prime number that I’m aware of is 2, so that means
This obviously has no solutions because is impossible, hence this case cannot hold.
Therefore, it must be the case that exactly one of is even.
Since was already shown to be impossible, then either
So both and/or .
Let’s first check if is satisfiable:
Hence is one solution.
Now, let’s check whether holds:
Therefore, only satisfies are all prime numbers.